answer：Okay, so I have this problem here: I need to figure out how to get the graph of y equals 3 to the power of (x plus 1) by shifting the graph of y equals 3 to the power of x. The options are shifting up, down, left, or right by one unit. Hmm, let me think about this step by step.First, I remember that when dealing with exponential functions like y = a^x, the graph has certain characteristics. For example, it increases rapidly as x increases and approaches zero as x becomes more negative. Now, if I have y = 3^x, that's the basic exponential function with base 3. Now, the question is about transforming this function into y = 3^(x+1). I think this involves some kind of shift. I recall that in function transformations, adding or subtracting inside the exponent affects the horizontal shift, while adding or subtracting outside affects the vertical shift. So, in this case, since the "+1" is inside the exponent, it should be a horizontal shift.But wait, I'm a bit confused about whether it's a shift to the left or to the right. I remember that for functions like y = f(x + h), if h is positive, it's a shift to the left by h units, and if h is negative, it's a shift to the right by |h| units. So, in this case, since we have y = 3^(x + 1), which can be thought of as y = f(x + 1) where f(x) = 3^x, that should be a shift to the left by 1 unit.Let me double-check that. If I consider a point on the original graph, say when x = 0, y = 3^0 = 1. On the transformed graph, y = 3^(0 + 1) = 3^1 = 3. So, the point (0,1) on the original graph moves to (0,3) on the transformed graph. Wait, that seems like a vertical shift up by 2 units, but that's not one of the options. Hmm, maybe I'm mixing things up.Wait, no. Actually, the shift is horizontal, so I should consider how the input x is affected. If I have y = 3^(x + 1), it's equivalent to replacing x with x + 1 in the original function. So, for any given y-value, the x-value needed is one less than it was before. That means the graph is shifted to the left by 1 unit because to get the same y-value, you need to input an x that's one less.Let me test this with another point. On the original graph, when x = 1, y = 3^1 = 3. On the transformed graph, to get y = 3, we need to solve 3^(x + 1) = 3. That means x + 1 = 1, so x = 0. So, the point (1,3) on the original graph moves to (0,3) on the transformed graph, which is indeed a shift to the left by 1 unit.I think I was confused earlier because I was looking at the y-values, but the shift is actually in the x-direction. So, the correct answer should be a shift to the left by 1 unit.

answer：Okay, so I have this problem here where I need to find the equation of the tangent line to the curve y = f(x) at the point Q(0, f(0)). The function f(x) is given as e^x plus another function g(x). They also tell me that the tangent line to y = g(x) at the point P(0, g(0)) is y = 2x + 1. Hmm, okay, let me try to break this down step by step.First, I remember that the equation of a tangent line at a point is given by y = f(a) + f’(a)(x - a), where f’(a) is the derivative of f at x = a. In this case, since we're looking at the point x = 0, the tangent line will be y = f(0) + f’(0)x.So, for the function g(x), they've given me the tangent line at x = 0 is y = 2x + 1. That means at x = 0, the value of g(0) is 1 because when x = 0, y = 2*0 + 1 = 1. So, g(0) = 1. Also, the slope of the tangent line at that point is 2, which means the derivative of g at x = 0, g’(0), is 2.Now, moving on to f(x). Since f(x) = e^x + g(x), I can find f(0) by plugging in x = 0. That would be e^0 + g(0). e^0 is 1, and we already found that g(0) is 1, so f(0) = 1 + 1 = 2.Next, I need to find the derivative of f(x) at x = 0, which is f’(0). The derivative of f(x) is the derivative of e^x plus the derivative of g(x). The derivative of e^x is e^x, so at x = 0, that's e^0 = 1. We already know that g’(0) is 2, so f’(0) = 1 + 2 = 3.Now that I have both f(0) and f’(0), I can write the equation of the tangent line. Using the point-slope form, it's y = f(0) + f’(0)x. Plugging in the values, that's y = 2 + 3x, or y = 3x + 2.Let me double-check to make sure I didn't make any mistakes. For g(x), the tangent line at x = 0 is y = 2x + 1, so g(0) = 1 and g’(0) = 2. For f(x), which is e^x + g(x), f(0) is e^0 + g(0) = 1 + 1 = 2. The derivative f’(x) is e^x + g’(x), so f’(0) is e^0 + g’(0) = 1 + 2 = 3. Therefore, the tangent line at x = 0 is y = 2 + 3x, which simplifies to y = 3x + 2. That seems correct.I think I've got it. The equation of the tangent line to y = f(x) at Q(0, f(0)) is y = 3x + 2.

answer：Okay, so I have this problem about Pete and his cousin Claire. It says that two years ago, Pete's age was three times Claire's age, and four years ago, Pete's age was four times Claire's age. I need to figure out in how many years their ages will have a ratio of 2:1. The options are A) 2, B) 4, C) 5, D) 6, E) 8.Hmm, let me start by defining some variables. Let me call Pete's current age P and Claire's current age C. That seems straightforward.Two years ago, Pete's age was P - 2, and Claire's age was C - 2. According to the problem, at that time, Pete's age was three times Claire's age. So I can write that as:P - 2 = 3*(C - 2)Similarly, four years ago, Pete's age was P - 4, and Claire's age was C - 4. The problem states that four years ago, Pete's age was four times Claire's age. So that gives me another equation:P - 4 = 4*(C - 4)Alright, so now I have two equations:1. P - 2 = 3*(C - 2)2. P - 4 = 4*(C - 4)I can solve these two equations to find P and C. Let me expand both equations first.Starting with the first equation:P - 2 = 3*(C - 2)P - 2 = 3C - 6P = 3C - 6 + 2P = 3C - 4Okay, so equation 1 simplifies to P = 3C - 4.Now, the second equation:P - 4 = 4*(C - 4)P - 4 = 4C - 16P = 4C - 16 + 4P = 4C - 12So equation 2 simplifies to P = 4C - 12.Now I have two expressions for P:1. P = 3C - 42. P = 4C - 12Since both equal P, I can set them equal to each other:3C - 4 = 4C - 12Let me solve for C.3C - 4 = 4C - 12Subtract 3C from both sides:-4 = C - 12Add 12 to both sides:8 = CSo Claire is currently 8 years old.Now, plug that back into one of the equations to find P. Let's use equation 1: P = 3C - 4P = 3*8 - 4P = 24 - 4P = 20So Pete is currently 20 years old, and Claire is 8.Wait, let me check if that makes sense with the original statements.Two years ago, Pete was 20 - 2 = 18, and Claire was 8 - 2 = 6. 18 is indeed three times 6. That works.Four years ago, Pete was 20 - 4 = 16, and Claire was 8 - 4 = 4. 16 is four times 4. That also works. Okay, so the current ages are correct.Now, the question is asking in how many years will Pete and Claire's ages have a ratio of 2:1.Let me denote the number of years from now as x. So in x years, Pete's age will be 20 + x, and Claire's age will be 8 + x.We want the ratio of their ages to be 2:1, so:(20 + x) / (8 + x) = 2 / 1Cross-multiplying:20 + x = 2*(8 + x)Let me solve this equation.20 + x = 16 + 2xSubtract x from both sides:20 = 16 + xSubtract 16 from both sides:4 = xSo x is 4. That means in 4 years, Pete's age will be twice Claire's age.Wait, let me check that.In 4 years, Pete will be 20 + 4 = 24, and Claire will be 8 + 4 = 12. 24 divided by 12 is indeed 2, so the ratio is 2:1. That works.So the answer should be 4 years, which is option B.But wait, let me make sure I didn't make any mistakes in my calculations.Starting from the beginning, I set up the equations correctly based on their ages two and four years ago. Solved for C and P, got 8 and 20, which checked out with the original conditions. Then set up the ratio equation, solved for x, got 4, which also checked out. So I think that's correct.I don't see any errors in my reasoning, so I think the answer is 4 years.**Final Answer**boxed{B}

answer：Okay, so I have this problem where I need to figure out how many numbers in a certain sequence are perfect squares. The sequence starts with 20142015, then 201402015, then 2014002015, and so on, each time adding an extra zero between the 4 and the 2. So, it's like 2014 followed by some zeros and then 2015. I need to check each of these numbers to see if any of them are perfect squares.First, I should probably write down the numbers to see the pattern clearly. The first few numbers are:1. 201420152. 2014020153. 20140020154. 201400020155. 2014000020156. ...Each subsequent number adds an extra zero between the 4 and the 2. So, the number of zeros increases by one each time. That means the length of the number increases by one each time as well. Now, I need to check if any of these numbers are perfect squares. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it's 4 squared. One way to check if a number is a perfect square is to take its square root and see if it's an integer. But these numbers are quite large, so calculating their square roots directly might be challenging. Maybe there's a pattern or a property I can use to determine if they can be perfect squares without computing the square roots.I remember that perfect squares have certain properties related to their digits. For example, the last digit of a perfect square can only be 0, 1, 4, 5, 6, or 9. Let me check the last digit of these numbers. Looking at the first number, 20142015, the last digit is 5. That's fine because 5 is a possible last digit for a perfect square. The second number, 201402015, also ends with 5. Same with the third, fourth, and so on. So, the last digit doesn't rule out the possibility of being a perfect square.Another property I recall is that the sum of the digits of a perfect square must be congruent to 0, 1, 4, 7, or 9 modulo 9. This is because of the divisibility rules related to 9. Let me calculate the sum of the digits for one of these numbers and see if it fits this criterion.Taking the first number, 20142015:- 2 + 0 + 1 + 4 + 2 + 0 + 1 + 5 = 1515 modulo 9 is 6 (since 15 ÷ 9 = 1 with a remainder of 6). 6 is not one of the allowed residues (0, 1, 4, 7, 9). Therefore, 20142015 cannot be a perfect square.Wait, but does this mean none of the numbers in the sequence can be perfect squares? Let me check another number to confirm. Take the second number, 201402015:- 2 + 0 + 1 + 4 + 0 + 2 + 0 + 1 + 5 = 15Again, the sum is 15, which modulo 9 is 6. Same result. Let me check the third number, 2014002015:- 2 + 0 + 1 + 4 + 0 + 0 + 2 + 0 + 1 + 5 = 15Still 15. It seems like every number in this sequence has a digit sum of 15 because the only digits that change are the zeros being added, which don't affect the sum. Therefore, every number in the sequence will have a digit sum of 15, which modulo 9 is 6. Since 6 isn't one of the allowed residues for perfect squares, none of these numbers can be perfect squares.I think that settles it. Since the digit sum of each number in the sequence is 15, and 15 modulo 9 is 6, which isn't a valid residue for perfect squares, none of these numbers can be perfect squares. Just to make sure I'm not missing something, I'll consider another property of perfect squares. For example, perfect squares can only end with certain two-digit combinations. The last two digits of a perfect square can be 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, or 96. Looking at the last two digits of our numbers: 15. 15 isn't in the list of possible two-digit endings for perfect squares. So, this further confirms that none of these numbers can be perfect squares.Another thought: maybe the numbers are too large to be perfect squares? Let's estimate the square roots. Take the first number, 20142015. Let's approximate its square root. I know that 4489 squared is 20,150,121 (since 4489^2 = (4500 - 11)^2 ≈ 4500^2 - 2*4500*11 + 11^2 = 20,250,000 - 99,000 + 121 = 20,151,121). Wait, that's actually larger than 20,142,015. Wait, maybe I miscalculated. Let me try 4489 squared:4489 * 4489. Let's compute 4489 * 4000 = 17,956,0004489 * 400 = 1,795,6004489 * 80 = 359,1204489 * 9 = 40,401Adding them up: 17,956,000 + 1,795,600 = 19,751,60019,751,600 + 359,120 = 20,110,72020,110,720 + 40,401 = 20,151,121So, 4489^2 = 20,151,121, which is larger than 20,142,015. Therefore, the square root of 20,142,015 is less than 4489. Let's try 4480^2:4480^2 = (4400 + 80)^2 = 4400^2 + 2*4400*80 + 80^2 = 19,360,000 + 704,000 + 6,400 = 20,070,400That's less than 20,142,015. So, the square root is between 4480 and 4489. Let's see how far 20,142,015 is from 20,070,400:20,142,015 - 20,070,400 = 71,615So, 4480^2 = 20,070,4004480 + x)^2 = 20,142,015Approximating x:(4480 + x)^2 ≈ 4480^2 + 2*4480*x = 20,070,400 + 8960xSet this equal to 20,142,015:20,070,400 + 8960x = 20,142,0158960x = 71,615x ≈ 71,615 / 8960 ≈ 7.99So, approximately 4480 + 8 = 4488. Let's check 4488^2:4488^2 = (4480 + 8)^2 = 4480^2 + 2*4480*8 + 8^2 = 20,070,400 + 71,680 + 64 = 20,142,144Wait, that's very close to 20,142,015. 20,142,144 is actually larger than 20,142,015 by 129. So, 4488^2 = 20,142,144Therefore, 4488^2 is 20,142,144, which is 129 more than 20,142,015. So, 20,142,015 is not a perfect square because it's between 4487^2 and 4488^2.Wait, let me check 4487^2:4487^2 = (4488 - 1)^2 = 4488^2 - 2*4488 + 1 = 20,142,144 - 8,976 + 1 = 20,133,169So, 4487^2 = 20,133,169, which is less than 20,142,015. Therefore, 20,142,015 is between 4487^2 and 4488^2, so it's not a perfect square.Similarly, the next number in the sequence is 201402015, which is 20,140,2015. Wait, actually, no, the numbers are 20142015, 201402015, etc. So, the second number is 20,140,2015? Wait, no, actually, the numbers are:1. 20142015 (8 digits)2. 201402015 (9 digits)3. 2014002015 (10 digits)4. 20140002015 (11 digits)5. 201400002015 (12 digits)6. ...So, each time, we're adding a zero, making the number longer by one digit each time. So, the first number is 8 digits, the second is 9 digits, and so on.Given that, the first number is 20,142,015, which we saw isn't a perfect square. The second number is 201,402,015. Let's check if that's a perfect square.Again, estimating the square root. Let's see, 14,000^2 = 196,000,00014,200^2 = (14,000 + 200)^2 = 14,000^2 + 2*14,000*200 + 200^2 = 196,000,000 + 5,600,000 + 40,000 = 201,640,000So, 14,200^2 = 201,640,000, which is larger than 201,402,015. So, the square root is less than 14,200.Let's try 14,190^2:14,190^2 = (14,200 - 10)^2 = 14,200^2 - 2*14,200*10 + 10^2 = 201,640,000 - 284,000 + 100 = 201,356,100That's less than 201,402,015. So, 14,190^2 = 201,356,100Difference: 201,402,015 - 201,356,100 = 45,915So, let's approximate how much more we need:Let x be the number such that (14,190 + x)^2 = 201,402,015Approximately, 2*14,190*x ≈ 45,915So, x ≈ 45,915 / (2*14,190) ≈ 45,915 / 28,380 ≈ 1.618So, approximately, 14,190 + 1.618 ≈ 14,191.618So, let's check 14,191^2:14,191^2 = (14,190 + 1)^2 = 14,190^2 + 2*14,190 + 1 = 201,356,100 + 28,380 + 1 = 201,384,481Still less than 201,402,015.Next, 14,192^2 = 14,191^2 + 2*14,191 + 1 = 201,384,481 + 28,382 + 1 = 201,412,864That's larger than 201,402,015. So, 14,192^2 = 201,412,864Therefore, 201,402,015 is between 14,191^2 and 14,192^2, so it's not a perfect square.Similarly, the third number is 2,014,002,015. Let's estimate its square root.Let's see, 44,890^2 = ?Wait, 44,890^2 = (44,000 + 890)^2 = 44,000^2 + 2*44,000*890 + 890^2 = 1,936,000,000 + 78,320,000 + 792,100 = 2,015,112,100That's larger than 2,014,002,015. So, the square root is less than 44,890.Let's try 44,800^2 = (44,000 + 800)^2 = 44,000^2 + 2*44,000*800 + 800^2 = 1,936,000,000 + 70,400,000 + 640,000 = 2,007,040,000That's less than 2,014,002,015. So, the square root is between 44,800 and 44,890.Difference: 2,014,002,015 - 2,007,040,000 = 6,962,015Approximating x such that (44,800 + x)^2 = 2,014,002,015Approximately, 2*44,800*x ≈ 6,962,015So, x ≈ 6,962,015 / (2*44,800) ≈ 6,962,015 / 89,600 ≈ 77.7So, approximately, 44,800 + 77.7 ≈ 44,877.7Let's check 44,877^2:44,877^2 = ?Well, 44,800^2 = 2,007,040,00044,877^2 = (44,800 + 77)^2 = 44,800^2 + 2*44,800*77 + 77^2 = 2,007,040,000 + 6,862,400 + 5,929 = 2,013,908,329That's still less than 2,014,002,015.Next, 44,878^2 = 44,877^2 + 2*44,877 + 1 = 2,013,908,329 + 89,754 + 1 = 2,014,000,084That's very close to 2,014,002,015. The difference is 2,014,002,015 - 2,014,000,084 = 1,931So, 44,878^2 = 2,014,000,084Then, 44,879^2 = 44,878^2 + 2*44,878 + 1 = 2,014,000,084 + 89,756 + 1 = 2,014,089,841That's way larger than 2,014,002,015. So, 2,014,002,015 is between 44,878^2 and 44,879^2, so it's not a perfect square.This pattern seems to be consistent. Each time, the number is not a perfect square because it falls between two consecutive squares. Also, the digit sum being 15, which modulo 9 is 6, which isn't a valid residue for perfect squares, further confirms that none of these numbers can be perfect squares.I think I've checked enough numbers to be confident that none of them are perfect squares. The properties related to digit sums and the position between consecutive squares both indicate that these numbers cannot be perfect squares.

answer：First, I recognize that the volume of a cylinder is calculated using the formula ( V = pi r^2 h ), where ( r ) is the radius and ( h ) is the height.Given that the original volume is 15 cubic feet, I can express this as:[ pi r^2 h = 15 ]When the radius is tripled, the new radius becomes ( 3r ). When the height is halved, the new height becomes ( frac{h}{2} ).Substituting these new dimensions into the volume formula, the new volume ( V' ) is:[ V' = pi (3r)^2 left(frac{h}{2}right) ]Simplifying this expression:[ V' = pi cdot 9r^2 cdot frac{h}{2} = frac{9}{2} pi r^2 h ]Since the original volume ( pi r^2 h ) is 15 cubic feet, the new volume becomes:[ V' = frac{9}{2} times 15 = frac{135}{2} = 67.5 , text{cubic feet} ]

answer：Okay, so I have this problem here: a complex number z satisfies z multiplied by i equals 1 plus i, and I need to find the magnitude of z, which is |z|. The options are A: sqrt(2)/2, B: 1, C: sqrt(2), and D: 2.Alright, let's start by recalling what a complex number is. A complex number is usually written as z = a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies i squared equals -1. The magnitude of a complex number, denoted |z|, is the distance from the origin to the point (a, b) in the complex plane, and it's calculated as sqrt(a^2 + b^2).So, the equation given is z * i = 1 + i. I need to solve for z first, and then find its magnitude.Let me write down the equation:z * i = 1 + iTo solve for z, I can divide both sides by i. So,z = (1 + i) / iHmm, dividing by i might be a bit tricky. Maybe I can simplify this expression by multiplying the numerator and the denominator by i to rationalize it. Let's try that:z = [(1 + i) * i] / (i * i)First, let's compute the numerator: (1 + i) * i. Using distributive property:1 * i + i * i = i + i^2But i squared is -1, so this becomes:i + (-1) = -1 + iNow, the denominator is i * i, which is i squared, which is -1.So, z = (-1 + i) / (-1)Dividing both terms in the numerator by -1:(-1)/(-1) + i/(-1) = 1 - iSo, z equals 1 - i.Now, to find the magnitude |z|, I use the formula sqrt(a^2 + b^2), where a is the real part and b is the imaginary part of z.In this case, z = 1 - i, so a = 1 and b = -1.Calculating the magnitude:|z| = sqrt(1^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2)So, the magnitude of z is sqrt(2), which corresponds to option C.Wait, let me double-check my steps to make sure I didn't make any mistakes.Starting with z * i = 1 + i.Divide both sides by i: z = (1 + i)/i.Multiply numerator and denominator by i: [(1 + i)*i]/(i*i) = (i + i^2)/(-1) = (i - 1)/(-1) = 1 - i.Yes, that seems correct.Then, |z| = sqrt(1^2 + (-1)^2) = sqrt(2). Yep, that checks out.So, I think option C is the correct answer.