answer：Alright, so I have this problem here about a bicyclist increasing their speed. Let me try to understand what's being asked. The problem says that the bicyclist first increases their speed by 30%, and then increases this new speed by another 10%. I need to find out what percent of the original speed this total increase is.Okay, let's break it down step by step. First, I need to figure out what happens when the bicyclist increases their speed by 30%. If the original speed is, say, S, then increasing it by 30% would mean adding 30% of S to itself. So, mathematically, that would be S plus 0.30 times S, which simplifies to 1.30 times S. Got that part.Now, the next part is where the bicyclist increases this new speed by another 10%. So, starting from 1.30S, increasing it by 10% would involve adding 10% of 1.30S to itself. That would be 1.30S plus 0.10 times 1.30S. Let me calculate that: 0.10 times 1.30S is 0.13S. So, adding that to 1.30S gives me 1.43S. Okay, so after both increases, the final speed is 1.43S.Now, I need to find out what percent of the original speed this total increase is. The total increase in speed is the final speed minus the original speed. So, that's 1.43S minus S, which equals 0.43S. To find out what percent this is of the original speed, I divide the increase by the original speed: 0.43S divided by S equals 0.43. To express this as a percentage, I multiply by 100, which gives me 43%.Wait a second, let me double-check that. If I start with S, increase it by 30% to get 1.30S, then increase that by 10% to get 1.43S, the total increase is indeed 0.43S, which is 43% of the original speed. That seems right.But just to make sure I didn't make a mistake, let me think about it another way. If the original speed is 100 units, increasing it by 30% would bring it to 130 units. Then increasing that by 10% would add 13 units, bringing the total to 143 units. The increase from 100 to 143 is 43 units, which is 43% of the original 100 units. Yep, that checks out.So, after both increases, the bicyclist's speed has increased by 43% compared to the original speed. I think that's the correct answer.

answer：First, I need to evaluate the function ( g(x) = x^3 ) at ( x = 3 ). This means I will calculate ( 3^3 ).Next, I will take the result from ( g(3) ) and substitute it into the function ( f(x) = 3x - 2 ). This involves multiplying the result by 3 and then subtracting 2 to find the final value of ( f(g(3)) ).

answer：Alright, so I have this problem here: points A, B, and C are on the same line l, and O is a point not on line l. The equation given is p times the vector OA plus q times the vector OB plus r times the vector OC equals zero. I need to find what p plus q plus r is.Hmm, okay. Let me try to visualize this. Points A, B, and C are colinear, meaning they lie on the same straight line l. Point O is somewhere else, not on that line. So, O is like a reference point from which we're measuring vectors to A, B, and C.The equation given is a vector equation: pOA + qOB + rOC = 0. So, this means that when you take p times the vector from O to A, plus q times the vector from O to B, plus r times the vector from O to C, you end up with the zero vector.I remember that when vectors are linearly dependent, you can express one vector as a combination of the others. Since A, B, and C are colinear, maybe there's a relationship between OA, OB, and OC.Wait, since A, B, and C are on the same line, the vectors OA, OB, and OC must be related somehow. Maybe I can express one of these vectors in terms of the others.Let me think. If A, B, and C are colinear, then the vectors AB and AC are scalar multiples of each other. That is, AB = λ AC for some scalar λ. So, AB = OB - OA, and AC = OC - OA. Therefore, OB - OA = λ(OC - OA).Let me write that down:OB - OA = λ(OC - OA)Expanding the right side:OB - OA = λ OC - λ OANow, let me rearrange terms:- OA + OB = λ OC - λ OABring all terms to one side:(-1 + λ) OA + OB - λ OC = 0So, that's an equation involving OA, OB, and OC. Comparing this with the given equation pOA + qOB + rOC = 0, I can see that:p = (-1 + λ)q = 1r = -λSo, p is (-1 + λ), q is 1, and r is -λ.Now, the question is asking for p + q + r. Let's compute that:p + q + r = (-1 + λ) + 1 + (-λ)Simplify the terms:-1 + λ + 1 - λThe λ terms cancel out: λ - λ = 0And -1 + 1 = 0So, p + q + r = 0Wait, that seems too straightforward. Did I miss something? Let me double-check.Starting from AB = λ AC, which gives OB - OA = λ(OC - OA). Then expanding and rearranging gives (-1 + λ) OA + OB - λ OC = 0. Comparing coefficients with pOA + qOB + rOC = 0, we get p = (-1 + λ), q = 1, r = -λ. Adding them up: (-1 + λ) + 1 + (-λ) = 0. Yeah, that seems correct.So, regardless of the value of λ, which depends on the specific positions of A, B, and C on line l, the sum p + q + r is always zero.I think that makes sense because the vectors OA, OB, and OC are linearly dependent due to the colinearity of A, B, and C. So, the coefficients in their linear combination must satisfy a certain condition, which in this case is that their sum is zero.Therefore, the answer should be 0.**Final Answer**The value of ( p + q + r ) is boxed{0}.

answer：Okay, so I have this function f(x) = 4 sin²x - 4 sinx sin2x + sin²2x, and I need to find its range. Hmm, the range means all the possible values that f(x) can take as x varies over all real numbers. Since it's a trigonometric function, I think I should try to simplify it using some trigonometric identities to make it easier to analyze.First, let me write down the function again:f(x) = 4 sin²x - 4 sinx sin2x + sin²2xI notice that there are terms with sinx and sin2x. Maybe I can express everything in terms of sinx or sin2x to simplify. Let me recall some trigonometric identities that might help.I remember that sin2x = 2 sinx cosx. Maybe I can substitute that into the equation to express everything in terms of sinx and cosx. Let's try that.So, sin2x = 2 sinx cosx. Therefore, sinx sin2x = sinx * 2 sinx cosx = 2 sin²x cosx.Similarly, sin²2x = (sin2x)² = (2 sinx cosx)² = 4 sin²x cos²x.Let me substitute these back into the function:f(x) = 4 sin²x - 4*(2 sin²x cosx) + 4 sin²x cos²xSimplify each term:First term: 4 sin²xSecond term: -4*(2 sin²x cosx) = -8 sin²x cosxThird term: 4 sin²x cos²xSo now, f(x) = 4 sin²x - 8 sin²x cosx + 4 sin²x cos²xHmm, I see that each term has a factor of 4 sin²x. Maybe I can factor that out to simplify further.Factoring out 4 sin²x:f(x) = 4 sin²x [1 - 2 cosx + cos²x]Now, inside the brackets, I have 1 - 2 cosx + cos²x. That looks familiar. It resembles the expansion of (1 - cosx)².Let me check: (1 - cosx)² = 1 - 2 cosx + cos²x. Yes, that's exactly what's inside the brackets.So, f(x) = 4 sin²x (1 - cosx)²Hmm, that's a nice simplification. So, f(x) is equal to 4 sin²x multiplied by (1 - cosx) squared.Now, I need to find the range of this function. Since f(x) is expressed in terms of sinx and cosx, which are both bounded functions, maybe I can find the maximum and minimum values of f(x).Let me think about how to approach this. Since f(x) is a product of sin²x and (1 - cosx)², both of which are non-negative, f(x) will always be non-negative. So, the minimum value of f(x) is 0. But I need to confirm if 0 is attainable.Looking at f(x) = 4 sin²x (1 - cosx)², it equals zero when either sinx = 0 or cosx = 1. Sinx = 0 when x is an integer multiple of π, and cosx = 1 when x is an even multiple of π. So, at x = 0, π, 2π, etc., f(x) = 0. Therefore, 0 is indeed in the range.Now, I need to find the maximum value of f(x). To find the maximum, perhaps I can express f(x) in terms of a single trigonometric function or use calculus to find critical points.Alternatively, since f(x) is non-negative, I can consider substituting t = sinx or t = cosx and express f(x) in terms of t, then find the maximum value.Let me try substituting t = cosx. Then, sin²x = 1 - t².So, f(x) = 4 (1 - t²) (1 - t)²Let me expand this expression:First, compute (1 - t)² = 1 - 2t + t²Then, f(x) = 4 (1 - t²)(1 - 2t + t²)Let me multiply (1 - t²) and (1 - 2t + t²):(1 - t²)(1 - 2t + t²) = 1*(1 - 2t + t²) - t²*(1 - 2t + t²)= (1 - 2t + t²) - (t² - 2t³ + t⁴)= 1 - 2t + t² - t² + 2t³ - t⁴= 1 - 2t + 0t² + 2t³ - t⁴= 1 - 2t + 2t³ - t⁴So, f(x) = 4*(1 - 2t + 2t³ - t⁴)Therefore, f(x) = 4 - 8t + 8t³ - 4t⁴Now, f(x) is expressed as a quartic function in terms of t, where t = cosx. Since t is the cosine of x, t ranges between -1 and 1. So, t ∈ [-1, 1].Therefore, to find the range of f(x), I can consider the function g(t) = 4 - 8t + 8t³ - 4t⁴, where t ∈ [-1, 1], and find its maximum and minimum values.So, now the problem reduces to finding the maximum and minimum of g(t) on the interval [-1, 1].To find the extrema, I can take the derivative of g(t) with respect to t, set it equal to zero, and solve for t. Then, evaluate g(t) at critical points and endpoints to find the maximum and minimum.Let me compute the derivative g’(t):g(t) = 4 - 8t + 8t³ - 4t⁴g’(t) = d/dt [4] - d/dt [8t] + d/dt [8t³] - d/dt [4t⁴]= 0 - 8 + 24t² - 16t³= -8 + 24t² - 16t³So, g’(t) = -16t³ + 24t² - 8To find critical points, set g’(t) = 0:-16t³ + 24t² - 8 = 0Let me factor out a common factor of -8:-8(2t³ - 3t² + 1) = 0So, 2t³ - 3t² + 1 = 0Now, I need to solve the cubic equation 2t³ - 3t² + 1 = 0.Let me try to factor this cubic equation. Maybe it has rational roots. By the Rational Root Theorem, possible rational roots are ±1, ±1/2.Let me test t = 1:2(1)^3 - 3(1)^2 + 1 = 2 - 3 + 1 = 0. So, t = 1 is a root.Therefore, (t - 1) is a factor. Let's perform polynomial division or factor it out.Divide 2t³ - 3t² + 1 by (t - 1):Using synthetic division:Coefficients: 2 | -3 | 0 | 1Wait, actually, the cubic is 2t³ - 3t² + 0t + 1.Divide by (t - 1):Set up synthetic division with root 1:1 | 2 -3 0 1Bring down the 2.Multiply 2 by 1: 2Add to next coefficient: -3 + 2 = -1Multiply -1 by 1: -1Add to next coefficient: 0 + (-1) = -1Multiply -1 by 1: -1Add to last coefficient: 1 + (-1) = 0So, the cubic factors as (t - 1)(2t² - t - 1)Now, factor the quadratic 2t² - t - 1.Looking for two numbers a and b such that a * b = 2*(-1) = -2 and a + b = -1.Hmm, factors of -2 that add up to -1: -2 and 1.So, split the middle term:2t² - 2t + t - 1Factor by grouping:(2t² - 2t) + (t - 1) = 2t(t - 1) + 1(t - 1) = (2t + 1)(t - 1)Therefore, the cubic factors as:(t - 1)(2t + 1)(t - 1) = (t - 1)^2 (2t + 1)So, the critical points are t = 1 (double root) and t = -1/2.So, critical points at t = 1 and t = -1/2.Now, we need to evaluate g(t) at critical points and endpoints of the interval t ∈ [-1, 1].So, the critical points are t = -1/2 and t = 1. The endpoints are t = -1 and t = 1.Wait, t = 1 is both a critical point and an endpoint. So, we need to evaluate g(t) at t = -1, t = -1/2, and t = 1.Let me compute g(t) at each of these points.First, t = -1:g(-1) = 4 - 8*(-1) + 8*(-1)^3 - 4*(-1)^4= 4 + 8 + (-8) - 4= (4 + 8) + (-8 - 4)= 12 - 12= 0Next, t = -1/2:g(-1/2) = 4 - 8*(-1/2) + 8*(-1/2)^3 - 4*(-1/2)^4Compute each term:4 is 4.-8*(-1/2) = 48*(-1/2)^3 = 8*(-1/8) = -1-4*(-1/2)^4 = -4*(1/16) = -1/4So, adding them up:4 + 4 - 1 - 1/4 = (4 + 4) + (-1 - 1/4) = 8 - 1.25 = 6.75Which is 27/4.Wait, 6.75 is 27/4? Let me check:27 divided by 4 is 6.75. Yes, correct.So, g(-1/2) = 27/4.Now, t = 1:g(1) = 4 - 8*(1) + 8*(1)^3 - 4*(1)^4= 4 - 8 + 8 - 4= (4 - 8) + (8 - 4)= (-4) + (4)= 0So, g(1) = 0.Therefore, the function g(t) attains its maximum value at t = -1/2, which is 27/4, and its minimum value at t = -1 and t = 1, which is 0.Therefore, the range of f(x) is [0, 27/4].Wait, but let me double-check my calculations because sometimes I might make a mistake.First, at t = -1:g(-1) = 4 - 8*(-1) + 8*(-1)^3 - 4*(-1)^4= 4 + 8 + (-8) - 4= 4 + 8 = 12; 12 - 8 = 4; 4 - 4 = 0. Correct.At t = -1/2:g(-1/2) = 4 - 8*(-1/2) + 8*(-1/2)^3 - 4*(-1/2)^4Compute each term:-8*(-1/2) = 48*(-1/2)^3 = 8*(-1/8) = -1-4*(-1/2)^4 = -4*(1/16) = -1/4So, 4 + 4 = 8; 8 - 1 = 7; 7 - 1/4 = 6.75, which is 27/4. Correct.At t = 1:g(1) = 4 - 8*1 + 8*1 - 4*1= 4 - 8 + 8 - 4= 0. Correct.So, yes, the maximum value is 27/4, and the minimum is 0.Therefore, the range of f(x) is from 0 to 27/4, inclusive.So, the range is [0, 27/4].**Final Answer**The range of the function is boxed{left[0, dfrac{27}{4}right]}.

answer：Okay, so I have this problem where there's an equilateral triangle ABC, and a point P inside it. From P, they drop perpendiculars to each of the sides: PQ to AB, PR to BC, and PS to CA. The lengths of these perpendiculars are given as PQ = x, PR = 2x, and PS = 3x. I need to find the side length AB in terms of x and radicals.Hmm, I remember something about equilateral triangles and the sum of the perpendicular distances from any interior point to the sides. Is it related to the height of the triangle? Let me think. Oh yeah, Viviani's theorem! It states that in an equilateral triangle, the sum of the perpendicular distances from any interior point to the three sides is equal to the height of the triangle.So, if I can find the height of triangle ABC, I can relate it to the given distances PQ, PR, and PS. Let me write that down.Given:- PQ = x- PR = 2x- PS = 3xAccording to Viviani's theorem:PQ + PR + PS = height of triangle ABCSo, substituting the given values:x + 2x + 3x = height6x = heightNow, I need to express the height in terms of the side length AB. Let me denote the side length as s. For an equilateral triangle, the height h can be calculated using the formula:h = (√3 / 2) * sSo, from Viviani's theorem, we have:6x = (√3 / 2) * sNow, I can solve for s:Multiply both sides by 2:12x = √3 * sThen, divide both sides by √3:s = (12x) / √3Hmm, I should rationalize the denominator. Multiply numerator and denominator by √3:s = (12x * √3) / (√3 * √3)s = (12x√3) / 3s = 4x√3So, the side length AB is 4√3 times x. Let me just double-check my steps to make sure I didn't make a mistake.1. Applied Viviani's theorem correctly: sum of perpendiculars equals height. That seems right.2. Calculated the height as 6x. Yes, x + 2x + 3x is 6x.3. Expressed height in terms of side length: h = (√3 / 2)s. Correct formula.4. Solved for s: 6x = (√3 / 2)s → 12x = √3 s → s = 12x / √3. Then rationalized to 4x√3. All steps look good.I think that's solid. So, the side length AB is 4√3 x.**Final Answer**The side length ( AB ) is boxed{4sqrt{3}x}.

answer：First, I need to determine how long it takes for each pipe to fill the tank individually. Pipe A can fill the tank in 32 minutes. Pipe B is 5 times faster than Pipe A, so it can fill the tank in 32 divided by 5, which is 6.4 minutes.Next, I'll calculate the filling rates of both pipes. Pipe A's rate is 1 tank per 32 minutes, which simplifies to 1/32 tanks per minute. Pipe B's rate is 1 tank per 6.4 minutes, which simplifies to 1/6.4 tanks per minute.To find the combined filling rate when both pipes are working together, I'll add their individual rates:1/32 + 1/6.4. Since 6.4 is equivalent to 32/5, the reciprocal is 5/32. Therefore, the combined rate is 1/32 + 5/32, which equals 6/32 or simplified to 3/16 tanks per minute.Finally, to determine how long it takes to fill the tank when both pipes are working together, I'll take the reciprocal of the combined rate:1 divided by (3/16), which equals 16/3 minutes, or approximately 5.33 minutes.