answer：Okay, so I'm trying to figure out this problem about a bank introducing a new term deposit service. The problem has two parts, and I need to tackle them one by one. Let me start by understanding what's given.First, it says that the volume of deposits is directly proportional to the deposit rate, with a proportional constant ( k ) where ( k > 0 ). That means if the deposit rate increases, the volume of deposits will also increase proportionally. So, mathematically, I can express the deposit volume ( g(x) ) as ( g(x) = kx ), where ( x ) is the deposit interest rate. The loan interest rate is set at 6%, which is 0.06 in decimal form. The bank expects to loan out all the deposits it receives. So, whatever amount people deposit, the bank will lend it out at 6% interest. Now, part (1) asks me to express the relationship between the deposit volume ( g(x) ) and the interest ( h(x) ) that the bank must pay to depositors as functions of the deposit interest rate ( x ). I already have ( g(x) = kx ). For the interest ( h(x) ), since the bank pays depositors an interest rate ( x ) on the deposits, the total interest paid would be the deposit rate multiplied by the deposit volume. So, ( h(x) = x cdot g(x) ). Substituting ( g(x) ) from above, that would be ( h(x) = x cdot kx = kx^2 ). So, for part (1), I have:- Deposit volume: ( g(x) = kx )- Interest paid: ( h(x) = kx^2 )Moving on to part (2), which asks for the deposit interest rate ( x ) that maximizes the bank's profit. To find this, I need to express the bank's profit as a function of ( x ) and then find the value of ( x ) that maximizes this profit. The bank's profit comes from the interest it earns from loans minus the interest it pays to depositors. The bank loans out all the deposits it receives, so the revenue from loans is the loan interest rate multiplied by the deposit volume. The interest paid to depositors is ( h(x) ) as calculated earlier. So, profit ( y ) can be expressed as:[ y = text{Revenue from loans} - text{Interest paid to depositors} ][ y = 0.06 cdot g(x) - h(x) ]Substituting ( g(x) = kx ) and ( h(x) = kx^2 ):[ y = 0.06 cdot kx - kx^2 ][ y = 0.06kx - kx^2 ]To find the maximum profit, I need to find the value of ( x ) that maximizes ( y ). This is a quadratic function in terms of ( x ), and since the coefficient of ( x^2 ) is negative (( -k )), the parabola opens downward, meaning the vertex is the maximum point.The general form of a quadratic function is ( y = ax^2 + bx + c ), and the vertex occurs at ( x = -frac{b}{2a} ). In this case, ( a = -k ) and ( b = 0.06k ). Plugging these into the vertex formula:[ x = -frac{0.06k}{2(-k)} ][ x = -frac{0.06k}{-2k} ][ x = frac{0.06}{2} ][ x = 0.03 ]So, the deposit interest rate should be set at 3% to maximize the bank's profit.Wait, let me double-check my calculations. I have ( y = 0.06kx - kx^2 ). Taking the derivative with respect to ( x ) to find the maximum:[ frac{dy}{dx} = 0.06k - 2kx ]Setting the derivative equal to zero for maximum:[ 0.06k - 2kx = 0 ][ 0.06k = 2kx ]Dividing both sides by ( k ) (since ( k > 0 )):[ 0.06 = 2x ][ x = 0.03 ]Yes, that confirms it. The maximum profit occurs at ( x = 0.03 ) or 3%. I think that makes sense because if the bank sets the deposit rate too high, it will pay more in interest, but if it's too low, it won't attract enough deposits. There's a sweet spot in between where the balance between attracting deposits and not paying too much in interest maximizes profit. So, putting it all together, the deposit interest rate should be set at 3% to achieve maximum profit.

answer：Okay, so I have this problem where I'm given a function ( g(x) = sqrt{x(1 - x)} ) defined on the interval [0, 1]. I need to find the area of the region enclosed by the graph of the function ( y = x g(x) ) and the x-axis. They mentioned that the area is ( frac{pi}{16} ), but I need to figure out how to get there.First, let me understand what ( g(x) ) looks like. The function ( g(x) = sqrt{x(1 - x)} ) is familiar—it's the equation of a semicircle. Specifically, if I square both sides, I get ( y^2 = x(1 - x) ), which simplifies to ( y^2 = -x^2 + x ). Rearranging terms, that's ( x^2 + y^2 - x = 0 ). Completing the square for the x-terms: ( x^2 - x + y^2 = 0 ). To complete the square for ( x^2 - x ), I take half of -1, which is -1/2, square it to get 1/4, and add and subtract that inside the equation: ( (x^2 - x + 1/4) + y^2 = 1/4 ). So, ( (x - 1/2)^2 + y^2 = (1/2)^2 ). That's the equation of a circle with center at (1/2, 0) and radius 1/2. Since ( g(x) ) is the square root, it's the upper half of this circle. So, ( g(x) ) is a semicircle sitting on the x-axis from 0 to 1.Now, the function ( y = x g(x) ) is given. Let me write that out: ( y = x sqrt{x(1 - x)} ). So, it's the product of x and the semicircle function. I need to find the area between this curve and the x-axis from x = 0 to x = 1.To find the area, I need to set up an integral of ( y = x sqrt{x(1 - x)} ) from 0 to 1. So, the area A is:[ A = int_{0}^{1} x sqrt{x(1 - x)} , dx ]Hmm, this integral looks a bit complicated. Maybe I can simplify it or use substitution. Let me see.First, let's rewrite the integrand:[ x sqrt{x(1 - x)} = x sqrt{x} sqrt{1 - x} = x^{3/2} (1 - x)^{1/2} ]So, the integral becomes:[ A = int_{0}^{1} x^{3/2} (1 - x)^{1/2} , dx ]This looks like a Beta function or maybe related to the Gamma function. I remember that integrals of the form ( int_{0}^{1} x^{c - 1} (1 - x)^{d - 1} , dx ) are Beta functions, denoted as ( B(c, d) ), and they relate to Gamma functions via ( B(c, d) = frac{Gamma(c) Gamma(d)}{Gamma(c + d)} ).In this case, c is 5/2 (since the exponent on x is 3/2, so c = 3/2 + 1 = 5/2) and d is 3/2 (since the exponent on (1 - x) is 1/2, so d = 1/2 + 1 = 3/2). So, the integral is:[ A = Bleft(frac{5}{2}, frac{3}{2}right) = frac{Gammaleft(frac{5}{2}right) Gammaleft(frac{3}{2}right)}{Gammaleft(frac{5}{2} + frac{3}{2}right)} = frac{Gammaleft(frac{5}{2}right) Gammaleft(frac{3}{2}right)}{Gamma(4)} ]I know that ( Gamma(n) = (n - 1)! ) for integer n, so ( Gamma(4) = 3! = 6 ). Now, for half-integers, the Gamma function is defined as:[ Gammaleft(frac{1}{2}right) = sqrt{pi} ][ Gammaleft(frac{3}{2}right) = frac{1}{2} sqrt{pi} ][ Gammaleft(frac{5}{2}right) = frac{3}{2} cdot frac{1}{2} sqrt{pi} = frac{3}{4} sqrt{pi} ]So, plugging these into the Beta function:[ A = frac{left(frac{3}{4} sqrt{pi}right) left(frac{1}{2} sqrt{pi}right)}{6} = frac{left(frac{3}{8} piright)}{6} = frac{3}{48} pi = frac{pi}{16} ]Wait, that's the answer they gave! So, the area is indeed ( frac{pi}{16} ). But let me double-check if I did everything correctly.Alternatively, maybe I can use substitution to solve the integral without invoking the Beta and Gamma functions. Let's try another approach.Let me set ( t = x ), so ( dt = dx ). Hmm, not helpful. Maybe a trigonometric substitution since we have a square root of a quadratic. Let me set ( x = sin^2 theta ). Then, ( dx = 2 sin theta cos theta dtheta ). Let's see:When ( x = 0 ), ( theta = 0 ). When ( x = 1 ), ( theta = frac{pi}{2} ).Substituting into the integral:[ A = int_{0}^{pi/2} sin^2 theta cdot sqrt{sin^2 theta (1 - sin^2 theta)} cdot 2 sin theta cos theta dtheta ]Simplify the square root:[ sqrt{sin^2 theta cos^2 theta} = sin theta cos theta ]So, the integral becomes:[ A = int_{0}^{pi/2} sin^2 theta cdot sin theta cos theta cdot 2 sin theta cos theta dtheta ]Simplify the terms:[ A = 2 int_{0}^{pi/2} sin^4 theta cos^2 theta dtheta ]Hmm, this seems more complicated. Maybe I should stick with the Beta function approach since it gave me the correct answer quickly.Alternatively, maybe I can use substitution ( u = x(1 - x) ). Let's try that.Let ( u = x(1 - x) ). Then, ( du = (1 - 2x) dx ). Hmm, but in the integral, I have ( x sqrt{u} dx ). Not sure if this substitution helps directly because of the x term.Wait, maybe I can express x in terms of u. Let's see:From ( u = x(1 - x) ), we can write ( u = x - x^2 ), which is a quadratic in x. Solving for x would involve the quadratic formula, but it might complicate things further.Alternatively, maybe I can split the integral into two parts, from 0 to 1/2 and from 1/2 to 1, and use symmetry.Let me consider the function ( y = x sqrt{x(1 - x)} ). If I reflect it around x = 1/2, maybe there's some symmetry.Let me substitute ( x = 1 - t ) in the integral from 1/2 to 1:When ( x = 1 - t ), ( dx = -dt ). So, the integral becomes:[ int_{1/2}^{1} x sqrt{x(1 - x)} dx = int_{0}^{1/2} (1 - t) sqrt{(1 - t) t} dt ]Which is:[ int_{0}^{1/2} (1 - t) sqrt{t(1 - t)} dt ]So, adding the two integrals:[ A = int_{0}^{1/2} x sqrt{x(1 - x)} dx + int_{0}^{1/2} (1 - t) sqrt{t(1 - t)} dt ]Let me change the variable back to x in the second integral:[ A = int_{0}^{1/2} x sqrt{x(1 - x)} dx + int_{0}^{1/2} (1 - x) sqrt{x(1 - x)} dx ]Combine the integrals:[ A = int_{0}^{1/2} [x + (1 - x)] sqrt{x(1 - x)} dx = int_{0}^{1/2} sqrt{x(1 - x)} dx ]Wait, that's interesting. So, the original integral from 0 to 1 is equal to twice the integral from 0 to 1/2 of ( sqrt{x(1 - x)} dx ).But ( sqrt{x(1 - x)} ) is the semicircle function, which we know the area of. The area under ( g(x) ) from 0 to 1 is half the area of the circle with radius 1/2, which is ( frac{1}{2} pi (1/2)^2 = frac{pi}{8} ).But wait, in this case, we have:[ A = 2 times int_{0}^{1/2} sqrt{x(1 - x)} dx ]But the total area under ( g(x) ) from 0 to 1 is ( frac{pi}{8} ), so the area from 0 to 1/2 is half of that, which is ( frac{pi}{16} ). Therefore, multiplying by 2 gives ( frac{pi}{8} ). Wait, that contradicts our earlier result.Wait, no. Let me clarify. The integral of ( g(x) ) from 0 to 1 is ( frac{pi}{8} ). So, the integral from 0 to 1/2 is half of that, which is ( frac{pi}{16} ). Therefore, the area A, which is twice the integral from 0 to 1/2 of ( sqrt{x(1 - x)} dx ), would be ( 2 times frac{pi}{16} = frac{pi}{8} ). But that's not matching the given answer of ( frac{pi}{16} ).Wait, I think I made a mistake in the substitution step. Let me go back.When I split the integral into two parts and made the substitution, I ended up with:[ A = int_{0}^{1/2} sqrt{x(1 - x)} dx ]But actually, the original integral A was:[ A = int_{0}^{1} x sqrt{x(1 - x)} dx ]After substitution, I found that:[ A = int_{0}^{1/2} sqrt{x(1 - x)} dx ]But wait, that can't be right because the original integrand was ( x sqrt{x(1 - x)} ), not just ( sqrt{x(1 - x)} ). So, perhaps I made a mistake in combining the integrals.Let me re-examine that step.Original integral:[ A = int_{0}^{1} x sqrt{x(1 - x)} dx ]Split into two parts:[ A = int_{0}^{1/2} x sqrt{x(1 - x)} dx + int_{1/2}^{1} x sqrt{x(1 - x)} dx ]Substitute ( x = 1 - t ) in the second integral:When ( x = 1 - t ), ( dx = -dt ), and when ( x = 1/2 ), ( t = 1/2 ). So, the second integral becomes:[ int_{1/2}^{0} (1 - t) sqrt{(1 - t) t} (-dt) = int_{0}^{1/2} (1 - t) sqrt{t(1 - t)} dt ]So, the total area A is:[ A = int_{0}^{1/2} x sqrt{x(1 - x)} dx + int_{0}^{1/2} (1 - x) sqrt{x(1 - x)} dx ]Combine the integrals:[ A = int_{0}^{1/2} [x + (1 - x)] sqrt{x(1 - x)} dx = int_{0}^{1/2} sqrt{x(1 - x)} dx ]Ah, so the x terms cancel out, leaving just the integral of ( sqrt{x(1 - x)} ) from 0 to 1/2.But the integral of ( sqrt{x(1 - x)} ) from 0 to 1 is the area of the semicircle, which is ( frac{pi}{8} ). Therefore, the integral from 0 to 1/2 is half of that, which is ( frac{pi}{16} ). So, A = ( frac{pi}{16} ).Wait, that makes sense now. So, by exploiting the symmetry of the function, I was able to simplify the integral and find that the area is indeed ( frac{pi}{16} ).Alternatively, using the Beta function approach, I also arrived at the same result, which is reassuring.So, to summarize, the area enclosed by ( y = x g(x) ) and the x-axis is ( frac{pi}{16} ).

answer：Okay, so I have this problem with two parts. Let me start by understanding what each part is asking.First, the function is given as f(x) = sqrt(3) sin x cos x + cos²x. I need to find the intervals where this function is strictly increasing. Hmm, okay. To find where a function is increasing, I remember that I need to look at its derivative. If the derivative is positive, the function is increasing there.So, step one: find f'(x). Let me compute that.f(x) = sqrt(3) sin x cos x + cos²xI can use the product rule for differentiation. Let's see:The derivative of sqrt(3) sin x cos x is sqrt(3) times (cos x * cos x + sin x * (-sin x)) because d/dx sin x = cos x and d/dx cos x = -sin x. So that would be sqrt(3)(cos²x - sin²x).Then, the derivative of cos²x is 2 cos x * (-sin x) = -2 sin x cos x.So putting it all together:f'(x) = sqrt(3)(cos²x - sin²x) - 2 sin x cos xHmm, that seems a bit complicated. Maybe I can simplify this expression using some trigonometric identities.I remember that cos²x - sin²x is equal to cos(2x), and 2 sin x cos x is equal to sin(2x). So let me rewrite f'(x):f'(x) = sqrt(3) cos(2x) - sin(2x)Okay, that's simpler. Now, I need to find where f'(x) > 0. So:sqrt(3) cos(2x) - sin(2x) > 0Hmm, this looks like a linear combination of sine and cosine. Maybe I can write this as a single sine or cosine function using the amplitude-phase form. I recall that a cos θ + b sin θ can be written as R cos(θ - φ), where R = sqrt(a² + b²) and tan φ = b/a.Wait, in this case, it's sqrt(3) cos(2x) - sin(2x). So, a = sqrt(3), b = -1.So, R = sqrt( (sqrt(3))² + (-1)² ) = sqrt(3 + 1) = sqrt(4) = 2.And tan φ = b/a = (-1)/sqrt(3) = -1/sqrt(3). So φ is the angle whose tangent is -1/sqrt(3). That would be -30 degrees or 330 degrees, which is equivalent to -π/6 radians.So, sqrt(3) cos(2x) - sin(2x) can be written as 2 cos(2x + π/6). Wait, let me check:Using the identity: a cos θ + b sin θ = R cos(θ - φ). But in our case, it's a cos θ - b sin θ, so maybe it's R cos(θ + φ). Let me verify.Alternatively, maybe it's better to write it as R sin(2x + φ). Let me try that.Wait, another approach: Let me factor out 2 from the expression:sqrt(3) cos(2x) - sin(2x) = 2 [ (sqrt(3)/2) cos(2x) - (1/2) sin(2x) ]Now, sqrt(3)/2 is cos(π/6) and 1/2 is sin(π/6). So:= 2 [ cos(π/6) cos(2x) - sin(π/6) sin(2x) ]Using the cosine addition formula: cos(A + B) = cos A cos B - sin A sin B.So, this becomes:= 2 cos(2x + π/6)Yes, that's correct. So, f'(x) = 2 cos(2x + π/6)So, f'(x) > 0 when cos(2x + π/6) > 0.Now, cosine is positive in the intervals (-π/2 + 2πk, π/2 + 2πk) for any integer k.So, 2x + π/6 must lie in (-π/2 + 2πk, π/2 + 2πk).Let me solve for x:-π/2 + 2πk < 2x + π/6 < π/2 + 2πkSubtract π/6 from all parts:-π/2 - π/6 + 2πk < 2x < π/2 - π/6 + 2πkSimplify the angles:-π/2 is -3π/6, so -3π/6 - π/6 = -4π/6 = -2π/3Similarly, π/2 is 3π/6, so 3π/6 - π/6 = 2π/6 = π/3So:-2π/3 + 2πk < 2x < π/3 + 2πkDivide all parts by 2:-π/3 + πk < x < π/6 + πkSo, the function f(x) is strictly increasing on the intervals (-π/3 + πk, π/6 + πk) for any integer k.That seems to be the answer for part (1).Now, moving on to part (2). It's about triangle ABC. Given f(A) = 1, vector AB · vector AC = 4, and BC = 2 sqrt(3). Need to find the lengths of sides AB and AC.First, let's recall that in triangle ABC, the sides opposite angles A, B, C are a, b, c respectively. So, BC is side a, AC is side b, AB is side c.Wait, actually, in standard notation, side a is opposite angle A, side b opposite angle B, and side c opposite angle C. So, if BC is given as 2 sqrt(3), then BC is side a, opposite angle A.Wait, but in the problem, it's given as BC = 2 sqrt(3). So, in triangle ABC, BC is the side opposite angle A. So, side a = BC = 2 sqrt(3).Also, f(A) = 1. So, f(A) = sqrt(3) sin A cos A + cos² A = 1.Let me compute f(A):f(A) = sqrt(3) sin A cos A + cos² AHmm, maybe I can express this in terms of double angles or something else.I know that sin(2A) = 2 sin A cos A, so sin A cos A = (1/2) sin(2A). Similarly, cos² A = (1 + cos(2A))/2.So, substituting:f(A) = sqrt(3) * (1/2) sin(2A) + (1 + cos(2A))/2= (sqrt(3)/2) sin(2A) + (1/2) + (1/2) cos(2A)Hmm, that looks like a combination of sine and cosine. Maybe I can write this as a single sine or cosine function.Let me write it as:f(A) = (sqrt(3)/2) sin(2A) + (1/2) cos(2A) + 1/2Wait, the first two terms can be combined into a single sine function.Let me consider the expression (sqrt(3)/2) sin(2A) + (1/2) cos(2A). This is of the form a sin θ + b cos θ, which can be written as R sin(θ + φ), where R = sqrt(a² + b²) and tan φ = b/a.Here, a = sqrt(3)/2, b = 1/2.So, R = sqrt( (sqrt(3)/2)^2 + (1/2)^2 ) = sqrt( 3/4 + 1/4 ) = sqrt(1) = 1.And tan φ = (1/2) / (sqrt(3)/2) = 1/sqrt(3), so φ = π/6.Therefore, (sqrt(3)/2) sin(2A) + (1/2) cos(2A) = sin(2A + π/6).So, f(A) = sin(2A + π/6) + 1/2.Given that f(A) = 1, so:sin(2A + π/6) + 1/2 = 1Subtract 1/2:sin(2A + π/6) = 1/2So, 2A + π/6 = π/6 + 2πk or 2A + π/6 = 5π/6 + 2πk, for integer k.Solving for A:Case 1: 2A + π/6 = π/6 + 2πkSubtract π/6:2A = 0 + 2πkSo, A = πkBut in a triangle, angles are between 0 and π, so A = 0 or π, which is not possible for a triangle. So, discard this case.Case 2: 2A + π/6 = 5π/6 + 2πkSubtract π/6:2A = 4π/6 + 2πk = 2π/3 + 2πkSo, A = π/3 + πkAgain, in a triangle, A must be between 0 and π, so A = π/3 or A = π/3 + π = 4π/3, but 4π/3 is more than π, so only A = π/3 is valid.So, angle A is π/3 radians, which is 60 degrees.Okay, so angle A is 60 degrees.Now, the next given is vector AB · vector AC = 4.I need to recall the dot product formula. The dot product of vectors AB and AC is equal to |AB| |AC| cos(theta), where theta is the angle between them.In this case, the angle between AB and AC is angle A, which we found to be π/3.So, vector AB · vector AC = |AB| |AC| cos(π/3) = 4We know that cos(π/3) = 1/2, so:|AB| |AC| * (1/2) = 4Multiply both sides by 2:|AB| |AC| = 8So, the product of sides AB and AC is 8.Also, we know that BC = 2 sqrt(3). Since BC is side a, opposite angle A, which is π/3.We can use the Law of Cosines here.Law of Cosines states that a² = b² + c² - 2bc cos AHere, a = BC = 2 sqrt(3), angle A = π/3, and sides b = AC, c = AB.So, plugging in:(2 sqrt(3))² = b² + c² - 2bc cos(π/3)Compute (2 sqrt(3))²: 4 * 3 = 12cos(π/3) = 1/2, so:12 = b² + c² - 2bc*(1/2)Simplify:12 = b² + c² - bcSo, we have two equations:1) b * c = 82) b² + c² - bc = 12We need to solve for b and c.Let me denote b and c as the sides AC and AB, respectively.Let me let b = AC, c = AB.So, equation 1: b c = 8Equation 2: b² + c² - b c = 12Let me substitute equation 1 into equation 2.From equation 1, b c = 8, so equation 2 becomes:b² + c² - 8 = 12So, b² + c² = 20Now, we have:b² + c² = 20andb c = 8We can use these to solve for b and c.Let me recall that (b + c)² = b² + 2 b c + c²We know b² + c² = 20 and b c = 8, so:(b + c)² = 20 + 2*8 = 20 + 16 = 36So, b + c = 6Similarly, (b - c)² = b² - 2 b c + c² = 20 - 16 = 4So, b - c = ±2Therefore, we have two possibilities:Case 1: b + c = 6 and b - c = 2Adding these equations:2b = 8 => b = 4Then, c = 6 - b = 2Case 2: b + c = 6 and b - c = -2Adding these equations:2b = 4 => b = 2Then, c = 6 - b = 4So, the solutions are (b, c) = (4, 2) or (2, 4)Therefore, the lengths of sides AB and AC are 2 and 4, or 4 and 2.Since AB and AC are just labels, it doesn't matter which is which, but in the context of the triangle, depending on the labeling, they could be either.So, the lengths are 2 and 4.Let me just verify if these satisfy the original equations.First, b c = 4 * 2 = 8, which matches equation 1.Second, b² + c² - b c = 16 + 4 - 8 = 12, which matches equation 2.Yes, that works.So, the lengths of sides AB and AC are 2 and 4.**Final Answer**(1) The function is strictly increasing on the intervals boxed{left[ kpi - frac{pi}{3}, kpi + frac{pi}{6} right]} for any integer k.(2) The lengths of sides AB and AC are boxed{2} and boxed{4} respectively.

answer：Alright, so I have this problem: "How many positive perfect squares less than (10^8) are multiples of 48?" Hmm, okay. Let me try to break this down step by step.First, I know that a perfect square is a number that's the square of an integer. So, numbers like 1, 4, 9, 16, and so on. The question is asking how many of these squares are less than (10^8) and also multiples of 48.Alright, so I need to find all perfect squares (N^2) such that (N^2 < 10^8) and (N^2) is divisible by 48. Let me write that down:1. (N^2 < 10^8)2. (48) divides (N^2), which means (N^2) is a multiple of 48.Okay, so first, let's figure out what it means for (N^2) to be a multiple of 48. I remember that if a number is a multiple of another, all the prime factors of the other must be present in the first number. So, let's factorize 48.48 can be factored into prime factors: (48 = 16 times 3 = 2^4 times 3). So, (N^2) must have at least (2^4) and (3) in its prime factors.But wait, (N^2) is a square, so all the exponents in its prime factorization must be even. That means if (N^2) has (2^4), that's fine because 4 is even. But it also needs to have at least one 3. However, since (N^2) is a square, the exponent of 3 must be even as well. So, (N^2) must have at least (3^2) to satisfy the divisibility by 3.So, putting that together, (N^2) must be divisible by (2^4 times 3^2). Let me compute that: (2^4 = 16) and (3^2 = 9), so (16 times 9 = 144). Therefore, (N^2) must be a multiple of 144.Wait, hold on. The original problem says multiples of 48, but I just concluded that (N^2) must be a multiple of 144. Is that correct? Let me double-check.Yes, because (N^2) needs to have at least (2^4) and (3^2) to be divisible by 48. So, 144 is the least common multiple of 48 and the square requirement. So, (N^2) must be a multiple of 144.So, now, I can rephrase the problem: How many perfect squares less than (10^8) are multiples of 144?That seems simpler. So, let me denote (N^2 = 144k^2), where (k) is a positive integer. Wait, no. If (N^2) is a multiple of 144, then (N) must be a multiple of the square root of 144, which is 12. So, (N = 12k), where (k) is a positive integer.Therefore, (N^2 = (12k)^2 = 144k^2). So, (144k^2 < 10^8). Let me write that inequality:(144k^2 < 10^8)I need to solve for (k). Let's divide both sides by 144:(k^2 < frac{10^8}{144})Compute (frac{10^8}{144}):First, (10^8 = 100,000,000).Divide that by 144:(100,000,000 ÷ 144). Let me compute that.Well, 144 × 694,444 = 100,000,000 approximately? Wait, let me do it step by step.144 × 694,444 = ?Wait, maybe it's easier to compute (100,000,000 ÷ 144).Divide numerator and denominator by 16: (100,000,000 ÷ 16 = 6,250,000), and 144 ÷ 16 = 9.So, now it's (6,250,000 ÷ 9 ≈ 694,444.444).So, (k^2 < 694,444.444). Therefore, (k < sqrt{694,444.444}).Compute (sqrt{694,444.444}).Well, (sqrt{694,444.444}) is approximately 833.333... because (833^2 = 694,489), which is a bit higher, and (833.333^2 ≈ 694,444.444).So, (k < 833.333). Since (k) must be an integer, the maximum value (k) can take is 833.Therefore, (k) can be any integer from 1 to 833, inclusive. So, the number of such (k) is 833.Wait, but hold on. Let me verify this because earlier I thought of 416, but now I'm getting 833. There must be a mistake somewhere.Wait, let's go back.I said that (N = 12k), so (N^2 = 144k^2). Then, (144k^2 < 10^8), so (k^2 < 10^8 / 144 ≈ 694,444.444), so (k < 833.333), so (k = 1) to (833), which is 833 numbers.But in the initial problem, the user had a different approach where they considered (N = 24k), leading to (k = 416). So, why the discrepancy?Wait, perhaps I made a mistake in the initial step. Let me think again.If (N^2) must be divisible by 48, which is (2^4 times 3). But since (N^2) is a square, the exponents in its prime factors must be even. So, for the factor of 2, which is (2^4), that's fine because 4 is even. But for the factor of 3, which is (3^1), since 1 is odd, we need to have at least (3^2) in (N^2). Therefore, (N^2) must be divisible by (2^4 times 3^2 = 16 times 9 = 144). So, that part seems correct.Therefore, (N) must be divisible by the square root of 144, which is 12. So, (N = 12k), leading to (N^2 = 144k^2). Then, (144k^2 < 10^8), so (k^2 < 10^8 / 144 ≈ 694,444.444), so (k < 833.333), so (k = 1) to (833), which is 833 numbers.But in the initial problem, the user considered (N = 24k), which would mean (N^2 = 576k^2), and then (k^2 < 10^8 / 576 ≈ 173,611.111), so (k < 416.43), so (k = 1) to (416), which is 416 numbers.So, why is there a difference? Which approach is correct?Wait, perhaps the initial approach was wrong because they considered (N = 24k), but actually, (N) only needs to be a multiple of 12, not necessarily 24. So, why did they take 24?Let me think. If (N) is a multiple of 12, then (N = 12k), and (N^2 = 144k^2). So, (N^2) is a multiple of 144, which is a multiple of 48. So, that should be sufficient.But in the initial problem, they took (N = 24k), which is a multiple of 24, which is a multiple of 12, but more restrictive. So, why did they do that?Wait, perhaps they thought that since 48 is (2^4 times 3), and (N^2) must have at least (2^4) and (3^2), so (N) must have at least (2^2) and (3^1), so (N) must be a multiple of (2^2 times 3 = 12). So, (N = 12k), which is correct.But in their solution, they took (N = 24k), which is (12 times 2k), so they are considering multiples of 24 instead of 12. So, why?Wait, maybe they made a mistake in their initial step. Let me see.They said: "Given that (48 = 16 cdot 3 = 2^4 cdot 3), a square (N^2) is divisible by 48 if it is divisible by (2^4 cdot 3^2 = 288). Thus, we look for (N^2 < 10^8) where (N) is a multiple of (sqrt{288} = 12sqrt{2})."Wait, hold on. They said that (N^2) must be divisible by (2^4 cdot 3^2 = 288), so (N) must be a multiple of (sqrt{288}). But (sqrt{288} = sqrt{144 times 2} = 12sqrt{2}), which is approximately 16.97.But (N) must be an integer, so they approximated it to 24, which is the next multiple of 12 that is also a multiple of (sqrt{2}). But that seems incorrect because (N) doesn't need to be a multiple of (sqrt{288}); rather, (N^2) needs to be a multiple of 288.Wait, actually, no. If (N^2) is a multiple of 288, then (N) must be a multiple of (sqrt{288}), but since (sqrt{288}) is not an integer, (N) must be a multiple of the smallest integer greater than (sqrt{288}) that makes (N^2) a multiple of 288.Wait, perhaps that's not the right way to think about it. Let me clarify.If (N^2) is divisible by 288, then (N) must be divisible by the square root of the largest square factor of 288. Let's factorize 288.288 = 16 × 18 = 16 × 2 × 9 = 2^5 × 3^2.So, the largest square factor is 16 × 9 = 144, which is 12^2. So, (N) must be divisible by 12, because (N^2) must be divisible by 144.Therefore, (N = 12k), and (N^2 = 144k^2). So, that's correct.But in the initial solution, they took (N = 24k), which is a multiple of 24, which is 2 × 12. So, why did they do that?Wait, perhaps they thought that since 288 is 2^5 × 3^2, then (N) must be divisible by 2^3 × 3, which is 24, because in (N^2), the exponent of 2 would be 6, which is more than 5, and the exponent of 3 would be 2, which is sufficient.But actually, (N) only needs to be divisible by 2^2 × 3, which is 12, because in (N^2), that would give 2^4 × 3^2, which is exactly 288. So, (N) needs to be a multiple of 12, not necessarily 24.Therefore, the initial solution was incorrect because they took (N = 24k), which is more restrictive than necessary, leading to a smaller count of 416 instead of the correct 833.Wait, but let me check again. If (N = 12k), then (N^2 = 144k^2). So, (144k^2 < 10^8), which gives (k^2 < 10^8 / 144 ≈ 694,444.444), so (k < 833.333), so (k = 1) to (833), which is 833 numbers.But in the initial problem, the user got 416 by taking (N = 24k), which leads to (k = 1) to (416). So, that's half the number.So, which one is correct? Let me think.If (N = 12k), then (N^2 = 144k^2), which is a multiple of 144, and hence a multiple of 48. So, all these squares are valid.But if (N = 24k), then (N^2 = 576k^2), which is a multiple of 576, which is a multiple of 48, but it's a stricter condition. So, the count would be smaller.Therefore, the correct count should be 833, not 416.Wait, but let me verify with an example. Let's take (k = 1). Then, (N = 12), (N^2 = 144), which is a multiple of 48. Good.(k = 2), (N = 24), (N^2 = 576), which is also a multiple of 48.But if I take (k = 3), (N = 36), (N^2 = 1296), which is also a multiple of 48.Wait, but in the initial solution, they took (N = 24k), which would correspond to (k = 1) being (N = 24), (k = 2) being (N = 48), etc. So, they are only counting every other multiple of 12, essentially.Therefore, their count was half of the actual count because they were only considering even multiples of 12, i.e., multiples of 24.So, the correct count should be 833, not 416.But wait, let me double-check the math.Compute (10^8 / 144):(10^8 = 100,000,000)Divide by 144:100,000,000 ÷ 144 ≈ 694,444.444So, (k^2 < 694,444.444), so (k < sqrt{694,444.444}).Compute (sqrt{694,444.444}):Well, 833^2 = 694,489, which is slightly more than 694,444.444, so (k) must be less than 833.333, so the maximum integer (k) is 833.Therefore, the number of such (k) is 833.But wait, in the initial solution, they had (k = 416). So, why the difference?Because they took (N = 24k), leading to (N^2 = 576k^2), and then (576k^2 < 10^8), so (k^2 < 10^8 / 576 ≈ 173,611.111), so (k < 416.43), so (k = 1) to (416).But this is only considering multiples of 24, which is a more restrictive condition than necessary. Therefore, their count is half of the actual count.So, the correct answer should be 833, not 416.Wait, but let me think again. Is (N = 12k) sufficient?Yes, because (N^2 = 144k^2) is a multiple of 144, which is a multiple of 48. So, all these squares are valid.Therefore, the number of such squares is 833.But wait, let me check with (k = 833):(N = 12 times 833 = 9996)(N^2 = 9996^2). Let's compute that:(9996^2 = (10,000 - 4)^2 = 100,000,000 - 2 times 10,000 times 4 + 4^2 = 100,000,000 - 80,000 + 16 = 99,920,016)Which is less than (10^8 = 100,000,000). So, that's valid.What about (k = 834):(N = 12 times 834 = 10,008)(N^2 = 10,008^2 = (10,000 + 8)^2 = 100,000,000 + 2 times 10,000 times 8 + 8^2 = 100,000,000 + 160,000 + 64 = 100,160,064)Which is greater than (10^8), so (k = 834) is invalid.Therefore, (k) can be from 1 to 833, inclusive, giving 833 perfect squares.So, the initial solution was incorrect because they took (N = 24k) instead of (N = 12k), leading to half the number of valid squares.Therefore, the correct answer should be 833.Wait, but let me think again. Is there a possibility that some of these (N = 12k) might not be perfect squares? No, because (N) is an integer, so (N^2) is a perfect square by definition.Therefore, all (N^2) where (N = 12k) and (k) from 1 to 833 are perfect squares less than (10^8) and multiples of 48.Hence, the correct count is 833.But wait, in the initial problem, the user had 416. So, perhaps they made a mistake in their reasoning.Alternatively, maybe I'm missing something. Let me think about the prime factorization again.Given that (N^2) must be divisible by 48, which is (2^4 times 3). Since (N^2) is a square, the exponents in its prime factors must be even. Therefore, for the factor of 2, which is (2^4), that's fine because 4 is even. For the factor of 3, which is (3^1), since 1 is odd, we need to have at least (3^2) in (N^2). Therefore, (N^2) must be divisible by (2^4 times 3^2 = 144).Therefore, (N) must be divisible by the square root of 144, which is 12. So, (N = 12k), leading to (N^2 = 144k^2).So, (144k^2 < 10^8), so (k^2 < 10^8 / 144 ≈ 694,444.444), so (k < 833.333), so (k = 1) to (833), which is 833 numbers.Therefore, the correct answer is 833.But wait, let me check with (k = 833):(N = 12 times 833 = 9996)(N^2 = 9996^2 = 99,920,016), which is less than (10^8).If (k = 834):(N = 12 times 834 = 10,008)(N^2 = 10,008^2 = 100,160,064), which is greater than (10^8).So, yes, (k) can be up to 833.Therefore, the number of perfect squares less than (10^8) that are multiples of 48 is 833.But wait, in the initial problem, the user had 416. So, why the discrepancy?Because they took (N = 24k), which is a multiple of 24, which is a multiple of 12, but more restrictive. So, they only counted every other multiple of 12, leading to half the number.Therefore, the correct answer is 833.Wait, but let me think again. Is there a possibility that some of these (N = 12k) might not be perfect squares? No, because (N) is an integer, so (N^2) is a perfect square by definition.Therefore, all (N^2) where (N = 12k) and (k) from 1 to 833 are perfect squares less than (10^8) and multiples of 48.Hence, the correct count is 833.But wait, let me think about the initial approach again. They said:"Given that (48 = 16 cdot 3 = 2^4 cdot 3), a square (N^2) is divisible by 48 if it is divisible by (2^4 cdot 3^2 = 288). Thus, we look for (N^2 < 10^8) where (N) is a multiple of (sqrt{288} = 12sqrt{2})."Wait, so they said that (N^2) must be divisible by 288, which is (2^5 times 3^2). Wait, no, 288 is (2^5 times 3^2), right? Because 288 = 2^5 × 3^2.But if (N^2) must be divisible by 288, which is (2^5 times 3^2), then (N) must be divisible by (2^{3} times 3^{1}), because in (N^2), the exponents would be doubled. So, (N) must be divisible by (2^{3} times 3^{1} = 8 times 3 = 24).Ah, so that's why they took (N = 24k). Because (N) must be divisible by 24 to ensure that (N^2) is divisible by (2^6 times 3^2), which is more than 288.Wait, but 288 is (2^5 times 3^2). So, to have (N^2) divisible by (2^5 times 3^2), (N) must be divisible by (2^{3} times 3^{1}), because (2^{2 times 3} = 2^6) and (3^{2 times 1} = 3^2). But (2^6) is more than (2^5), so it's sufficient.But actually, (N^2) only needs to have at least (2^5) and (3^2). So, (N) needs to have at least (2^{3}) and (3^{1}), because (2^{2 times 3} = 2^6) and (3^{2 times 1} = 3^2). So, (N) must be divisible by (2^3 times 3 = 24).Therefore, (N = 24k), leading to (N^2 = 576k^2), which is divisible by 576, which is (2^6 times 3^2), which is more than 288.But wait, 576 is a multiple of 288, so it's sufficient, but it's more restrictive. Therefore, the count would be smaller.But in reality, (N^2) only needs to be divisible by 288, not necessarily 576. So, (N) only needs to be divisible by the square root of 288, but since 288 is not a perfect square, we take the smallest integer (N) such that (N^2) is divisible by 288.Wait, but 288 is (2^5 times 3^2). So, to have (N^2) divisible by (2^5 times 3^2), (N) must be divisible by (2^{3} times 3^{1}), because (2^{2 times 3} = 2^6) and (3^{2 times 1} = 3^2). So, (N) must be divisible by 24.Therefore, (N = 24k), leading to (N^2 = 576k^2), which is divisible by 576, which is a multiple of 288.But wait, 576 is a multiple of 288, so (N^2) being divisible by 576 implies it's divisible by 288, but the converse isn't true. So, if we take (N = 24k), we are ensuring that (N^2) is divisible by 576, which is more than required. Therefore, we are excluding some valid (N) that would make (N^2) divisible by 288 but not by 576.Wait, but is that possible? Let me think.If (N) is divisible by 12 but not by 24, then (N = 12k) where (k) is odd. Then, (N^2 = 144k^2), which is divisible by 144, but not necessarily by 288.Wait, because 144k^2 is divisible by 144, but to be divisible by 288, it needs to have an additional factor of 2. So, if (k) is even, then (k^2) is divisible by 4, so (144k^2) is divisible by 576, which is 144 × 4. But if (k) is odd, then (k^2) is odd, so (144k^2) is divisible by 144 but not by 288.Therefore, if (k) is even, (N^2) is divisible by 576, which is a multiple of 288. If (k) is odd, (N^2) is divisible by 144 but not by 288.But wait, the original problem is asking for perfect squares that are multiples of 48, not 288. So, do we need (N^2) to be divisible by 48 or 288?Wait, the problem is multiples of 48. So, (N^2) needs to be divisible by 48, which is (2^4 times 3). Since (N^2) is a square, it must have even exponents in its prime factors. Therefore, (N^2) must have at least (2^4) and (3^2), which is 144. So, (N^2) must be divisible by 144, not necessarily 288.Therefore, (N) must be divisible by 12, not necessarily 24.Therefore, the initial approach of taking (N = 24k) was incorrect because it was ensuring divisibility by 576, which is more than required. Therefore, the correct count is 833, not 416.So, in conclusion, the number of positive perfect squares less than (10^8) that are multiples of 48 is 833.

answer：Okay, so I have this problem here: Katherine has 34 bottle caps. 8 are eaten by a hippopotamus. How many bottle caps will Katherine have? Alright, let's break this down step by step.First, I need to understand what the problem is asking. It seems pretty straightforward. Katherine starts with a certain number of bottle caps, and then some are eaten by a hippo. I need to find out how many she has left after that happens.So, the initial number of bottle caps is 34. That's clear. Then, 8 are eaten by a hippopotamus. Hmm, that's interesting. I wonder how a hippo would eat bottle caps, but maybe that's just part of the story to make it more engaging.Now, to find out how many bottle caps Katherine has left, I need to subtract the number of bottle caps eaten by the hippo from the original number she had. So, mathematically, that would be 34 minus 8.Let me write that down: 34 - 8. Okay, now I need to perform this subtraction. I can do this in my head, but maybe writing it down would help me visualize it better.Starting with 34, if I take away 8, I can think of it as subtracting 10 and then adding back 2, because 10 is easier to subtract mentally. So, 34 minus 10 is 24, and then adding back 2 gives me 26. That seems right.Alternatively, I could break it down digit by digit. The number 34 has a 3 in the tens place and a 4 in the ones place. Subtracting 8 from the ones place: 4 minus 8. Wait, 4 is less than 8, so I need to borrow from the tens place. That means the 3 in the tens place becomes 2, and the 4 in the ones place becomes 14. Now, 14 minus 8 is 6, and the tens place is now 2. So, putting it together, I get 26.Both methods give me the same result, which is reassuring. So, after the hippo eats 8 bottle caps, Katherine has 26 left.I should double-check my work to make sure I didn't make any mistakes. Let me add 26 and 8 to see if I get back to 34. 26 plus 8 is 34. Perfect, that confirms that my subtraction was correct.Another way to think about it is using a number line. Starting at 34, if I move back 8 units, where do I land? Let's count backwards: 34, 33, 32, 31, 30, 29, 28, 27, 26. Yep, that's 8 steps back, landing on 26.I could also use objects to visualize this. If I have 34 small items, like coins or beads, and I remove 8 of them, I can count what's left. That hands-on approach might help solidify the concept, especially for someone just learning subtraction.It's also worth noting that subtraction is the inverse of addition. So, if I have 34 and I take away 8, I'm essentially finding out what number, when added to 8, gives me 34. That number is 26 because 26 plus 8 equals 34.Thinking about place value again, 34 is composed of 3 tens and 4 ones. When I subtract 8, which is less than 10, I only affect the ones place. Since 4 ones aren't enough to subtract 8, I need to borrow from the tens place, turning one ten into 10 ones. That gives me 14 ones, and subtracting 8 leaves me with 6 ones. The tens place, which was originally 3, becomes 2 after borrowing. So, 2 tens and 6 ones make 26.I could also use mental math strategies, like counting up from 8 to 34. Starting at 8, how many do I need to add to reach 34? Let's see: 8 plus 2 is 10, and then from 10 to 34 is 24. So, 2 plus 24 is 26. That's another way to arrive at the same answer.Visualizing this with base-10 blocks might help. If I have 3 tens rods and 4 unit cubes, and I take away 8 unit cubes, I realize I don't have enough units. So, I exchange one ten rod for 10 unit cubes, leaving me with 2 tens rods and 14 unit cubes. Now, I can easily take away 8 units, leaving me with 6 units. So, 2 tens and 6 units equal 26.It's also useful to understand why subtraction works this way. When we subtract, we're essentially finding the difference between two numbers. In this case, the difference between 34 and 8 is 26, which tells us how many bottle caps Katherine has left after the hippo ate some.I can also think about this problem in terms of real-life scenarios. If I have 34 candies and I give 8 to a friend, how many do I have left? It's the same concept. Subtracting the number given away from the total gives me the remaining amount.Another perspective is using money. If I have 34 and I spend 8, how much do I have left? Again, it's a subtraction problem. 34 minus 8 equals 26. This helps in understanding the practical application of subtraction in everyday situations.I could also use algebra to represent this problem. Let's let x be the number of bottle caps Katherine has left. So, the equation would be:34 - 8 = xSolving for x gives:x = 26This shows that algebra can be used to represent and solve simple subtraction problems.It's also important to understand the properties of subtraction. Subtraction is not commutative, meaning that the order of the numbers matters. For example, 34 minus 8 is not the same as 8 minus 34. The first gives a positive result, while the second would give a negative result, which doesn't make sense in this context since you can't have a negative number of bottle caps.Additionally, subtraction is the inverse of addition. So, if I add 8 to 26, I should get back to 34, which I do. This inverse relationship is fundamental in arithmetic and helps in checking the correctness of calculations.Thinking about the number bonds, 34 can be broken down into 26 and 8. This helps in understanding how numbers relate to each other and can be useful in more complex mathematical operations down the line.I could also use a calculator to verify my answer. Typing in 34 minus 8 gives me 26, which matches my previous calculations. This is a good way to ensure that my mental math and manual calculations are accurate.In summary, by using various methods such as mental subtraction, borrowing in subtraction, number lines, real-life analogies, algebra, and calculators, I've consistently arrived at the answer of 26. This thorough exploration helps solidify my understanding of the concept and ensures that my solution is correct.

answer：Let ( q ) be the total number of people initially in the choir. Since 60% of the choir are girls, the initial number of girls is ( 0.6q ).After four girls leave and six boys join while two boys leave, the total number of people changes. The number of girls becomes ( 0.6q - 4 ), and the number of boys becomes ( q - 0.6q + 6 - 2 ).After the change, 50% of the choir are girls. Therefore, the equation becomes:[frac{0.6q - 4}{q} = 0.5]Solving the equation:[0.6q - 4 = 0.5q][0.6q - 0.5q = 4][0.1q = 4][q = frac{4}{0.1} = 40]The initial number of girls is:[0.6 times 40 = 24]The initial number of girls in the choir was ( 24 ).