💡First, I recognize that an equilateral triangle has all three sides of equal length. Given that the perimeter of the equilateral triangle is 60, I can determine the length of each side by dividing the perimeter by 3.Next, I understand that one side of this equilateral triangle is also a side of an isosceles triangle with a perimeter of 70. An isosceles triangle has two sides of equal length (the legs) and a third side (the base). Since one of the legs is the same as the side of the equilateral triangle, I know the length of the legs.To find the length of the base of the isosceles triangle, I subtract the combined length of the two legs from the total perimeter of the isosceles triangle.

💡Okay, so I have this problem here: The graph of the function y = a^(x-4) + 1, where a is greater than 0 and not equal to 1, always passes through a fixed point P. If point P lies on the graph of the power function f(x), then what is f(x)?Hmm, let me try to figure this out step by step. First, I remember that exponential functions like y = a^x have certain properties. For example, they always pass through (0,1) because any number to the power of 0 is 1. But in this case, the function is shifted. It's y = a^(x-4) + 1. So, it's shifted 4 units to the right and 1 unit up.Wait, so if it's shifted 4 units to the right, that means the point that was originally at (0,1) for y = a^x is now at (4,1) for y = a^(x-4). Then, since it's also shifted up by 1, the point becomes (4,2). So, does that mean the fixed point P is (4,2)? That seems right because no matter what value a takes (as long as a > 0 and a ≠ 1), the function will always pass through (4,2). Let me check that.If I plug x = 4 into the function, y = a^(4-4) + 1 = a^0 + 1 = 1 + 1 = 2. Yep, so regardless of a, when x is 4, y is always 2. So, P is definitely (4,2).Now, the problem says that this point P lies on the graph of the power function f(x). So, f(x) is a power function, which generally looks like f(x) = x^k, where k is some constant. So, I need to find k such that when x is 4, f(x) is 2.Let me write that down: f(4) = 2. Since f(x) = x^k, then f(4) = 4^k = 2. So, I need to solve for k in the equation 4^k = 2.Hmm, how do I solve this? Well, I know that 4 is 2 squared, so 4 = 2^2. Therefore, 4^k = (2^2)^k = 2^(2k). So, 2^(2k) = 2^1. Since the bases are the same, the exponents must be equal. So, 2k = 1. Solving for k, I get k = 1/2.So, that means the power function f(x) is x^(1/2), which is the same as the square root of x. Therefore, f(x) = sqrt(x).Let me just double-check my steps to make sure I didn't make any mistakes. First, I found the fixed point P by plugging x = 4 into the function y = a^(x-4) + 1, which gave me y = 2. Then, I used the fact that P lies on the power function f(x) = x^k, so plugging in x = 4 and f(x) = 2, I got 4^k = 2. By expressing 4 as 2^2, I rewrote the equation as 2^(2k) = 2^1, leading to 2k = 1 and k = 1/2. Therefore, f(x) = x^(1/2) or sqrt(x). That seems correct.I don't think I made any mistakes here. The logic flows logically from one step to the next, and each part checks out. So, I'm confident that the answer is f(x) = sqrt(x).**Final Answer**The power function is boxed{sqrt{x}}.

💡Okay, so I have this math problem here: "9 minus what number ÷ 1/3 + 3 equals 3?" Hmm, let's try to make sense of it step by step. First, I need to figure out what the unknown number is. Let's call that number "x." So, the equation becomes:9 - (x ÷ (1/3)) + 3 = 3Alright, now I need to solve for x. I remember that dividing by a fraction is the same as multiplying by its reciprocal. So, x ÷ (1/3) is the same as x multiplied by 3. Let me rewrite the equation with that in mind:9 - (x * 3) + 3 = 3Okay, so now the equation is 9 minus 3x plus 3 equals 3. Next, I should combine the constant terms on the left side. 9 plus 3 is 12, so the equation simplifies to:12 - 3x = 3Now, I need to isolate x. To do that, I'll subtract 12 from both sides of the equation:12 - 3x - 12 = 3 - 12This simplifies to:-3x = -9Alright, now I have -3x equals -9. To solve for x, I'll divide both sides by -3:(-3x)/(-3) = (-9)/(-3)This simplifies to:x = 3So, the unknown number is 3. Let me double-check to make sure I did everything correctly. If I plug x = 3 back into the original equation:9 - (3 ÷ (1/3)) + 3 = 3First, 3 divided by (1/3) is 9, so:9 - 9 + 3 = 3Which simplifies to:0 + 3 = 3And that's true, so x = 3 is indeed the correct answer.

💡First, let's define the number of marbles Carl initially took out as ( x ).Carl loses half of these marbles, which is ( frac{x}{2} ). He then takes out an additional 10 marbles, so the total number of marbles he has outside the jar is ( frac{x}{2} + 10 ).His mother brings him 25 new marbles, making the total number of marbles he has ( frac{x}{2} + 10 + 25 ).According to the problem, Carl will put 41 marbles back into the jar. Therefore, we set up the equation:[frac{x}{2} + 10 + 25 = 41]Simplifying the equation:[frac{x}{2} = 6]Multiplying both sides by 2 to solve for ( x ):[x = 12]Thus, Carl initially took out (boxed{12}) marbles to play the game.

💡Okay, so I have this problem where I need to compute the sum of sine squared terms at specific angles. The angles are 3 degrees, 9 degrees, 15 degrees, and so on, up to 177 degrees. Hmm, that seems like a sequence of angles increasing by 6 degrees each time. Let me verify that.Starting from 3 degrees, adding 6 degrees each time: 3, 9, 15, 21, ..., 177. Yes, that's correct. So, how many terms are there in this sequence? Let me calculate that.The first term is 3 degrees, and each subsequent term increases by 6 degrees. The last term is 177 degrees. To find the number of terms, I can use the formula for the nth term of an arithmetic sequence:a_n = a_1 + (n - 1)dWhere a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms.Plugging in the values:177 = 3 + (n - 1) * 6Subtract 3 from both sides:174 = (n - 1) * 6Divide both sides by 6:29 = n - 1Add 1 to both sides:n = 30So, there are 30 terms in this sum. That's good to know because it might help later on.Now, the sum I need to compute is:sin²(3°) + sin²(9°) + sin²(15°) + ... + sin²(177°)I remember that there's a trigonometric identity that can help simplify sin²(x). The double-angle identity for cosine states that:cos(2x) = 1 - 2sin²(x)So, solving for sin²(x):sin²(x) = (1 - cos(2x)) / 2That's useful. If I apply this identity to each term in the sum, I can rewrite the entire sum in terms of cosines. Let me try that.Each term sin²(3°), sin²(9°), etc., becomes (1 - cos(6°))/2, (1 - cos(18°))/2, and so on. So, the entire sum becomes:Sum = [ (1 - cos(6°))/2 + (1 - cos(18°))/2 + (1 - cos(30°))/2 + ... + (1 - cos(354°))/2 ]Since each term is divided by 2, I can factor that out:Sum = (1/2) * [ (1 - cos(6°)) + (1 - cos(18°)) + (1 - cos(30°)) + ... + (1 - cos(354°)) ]Now, let's separate the constants and the cosine terms:Sum = (1/2) * [ (1 + 1 + 1 + ... + 1) - (cos(6°) + cos(18°) + cos(30°) + ... + cos(354°)) ]There are 30 terms in the sum, so the sum of the constants is 30. Therefore:Sum = (1/2) * [ 30 - (cos(6°) + cos(18°) + cos(30°) + ... + cos(354°)) ]So, now the problem reduces to computing the sum of cosines: cos(6°) + cos(18°) + cos(30°) + ... + cos(354°). Let me denote this sum as S.S = cos(6°) + cos(18°) + cos(30°) + ... + cos(354°)I need to find the value of S. Hmm, how can I compute this sum? I remember that sums of cosines can sometimes be evaluated using complex numbers or by recognizing them as parts of geometric series.Let me consider using complex exponentials. Euler's formula tells us that:cos(θ) = Re(e^(iθ))So, the sum S can be written as the real part of the sum of complex exponentials:S = Re( e^(i6°) + e^(i18°) + e^(i30°) + ... + e^(i354°) )Let me denote this complex sum as W:W = e^(i6°) + e^(i18°) + e^(i30°) + ... + e^(i354°)So, S = Re(W). Now, let's see if we can compute W.Notice that the angles in W are 6°, 18°, 30°, ..., 354°, which is an arithmetic sequence with a common difference of 12°. Wait, 18° - 6° = 12°, 30° - 18° = 12°, and so on. So, the angles are increasing by 12° each time.But wait, in the original sum, the angles were increasing by 6°, but when we applied the double-angle identity, we got 2x, so 3° became 6°, 9° became 18°, etc., so the step between the arguments of the cosine terms is 12°. That makes sense.So, W is a geometric series where each term is multiplied by e^(i12°). Let me verify that.The first term is e^(i6°), the second term is e^(i18°) = e^(i6°) * e^(i12°), the third term is e^(i30°) = e^(i18°) * e^(i12°), and so on. Yes, so W is a geometric series with the first term a = e^(i6°) and common ratio r = e^(i12°).The formula for the sum of a geometric series is:Sum = a * (1 - r^n) / (1 - r)Where n is the number of terms. Earlier, we found that there are 30 terms in the original sum, so n = 30.Therefore, W can be written as:W = e^(i6°) * (1 - (e^(i12°))^30) / (1 - e^(i12°))Simplify the exponent in the numerator:(e^(i12°))^30 = e^(i12°*30) = e^(i360°) = e^(i2π) = 1Because 360° is equivalent to 2π radians, and e^(i2π) = 1.So, substituting back:W = e^(i6°) * (1 - 1) / (1 - e^(i12°)) = 0 / (1 - e^(i12°)) = 0Wait, that can't be right. If W is zero, then S = Re(W) = 0. But let me double-check my calculations.Wait, the numerator is (1 - e^(i360°)) which is (1 - 1) = 0. So, W = 0. Therefore, S = Re(0) = 0.Hmm, so the sum of the cosines is zero. That simplifies things.Therefore, going back to the original sum:Sum = (1/2) * [30 - 0] = (1/2) * 30 = 15Wait, but hold on. In the initial steps, I thought the sum was 30, but now I'm getting 15. Did I make a mistake?Wait, let's go back step by step.Original sum: sin²(3°) + sin²(9°) + ... + sin²(177°)Number of terms: 30Using the identity: sin²(x) = (1 - cos(2x))/2So, each term becomes (1 - cos(2x))/2, so the sum becomes:Sum = sum_{k=1}^{30} [ (1 - cos(2x_k)) / 2 ] where x_k = 3°, 9°, ..., 177°Which is equivalent to:Sum = (1/2) * [ sum_{k=1}^{30} 1 - sum_{k=1}^{30} cos(2x_k) ]Sum of 1's: 30Sum of cos(2x_k): sum_{k=1}^{30} cos(6°, 18°, 30°, ..., 354°) which we denoted as S.We found that S = 0, so:Sum = (1/2) * [30 - 0] = 15Wait, so the sum should be 15, not 30. But in the initial problem, the user wrote:Compute the sum sin²3° + sin²9° + ... + sin²177°.But in the assistant's solution, it was concluded as 30. Hmm, that seems conflicting.Wait, let me check the assistant's solution again.The assistant wrote:The sum is equivalent to 30 - (1/2)(0) = 30.But according to my calculation, it should be (1/2)(30 - 0) = 15.Wait, perhaps the assistant made a mistake in the calculation.Wait, in the assistant's solution, they wrote:The sum is equivalent to 30 - (1/2)(0) = 30.But according to the transformation, it should be (1/2)(30 - sum(cos terms)).So, if sum(cos terms) = 0, then the sum is (1/2)(30) = 15.Therefore, the correct answer should be 15, not 30.But in the assistant's solution, they wrote 30. So, perhaps the assistant made an error.Alternatively, maybe I made a mistake in my calculation.Wait, let me double-check.Original sum: sum_{k=1}^{30} sin²(x_k), where x_k = 3°, 9°, ..., 177°Using sin²(x) = (1 - cos(2x))/2, so sum becomes:sum = (1/2) * sum_{k=1}^{30} (1 - cos(2x_k)) = (1/2)(30 - sum_{k=1}^{30} cos(2x_k))So, sum = 15 - (1/2) sum_{k=1}^{30} cos(2x_k)Now, 2x_k are 6°, 18°, 30°, ..., 354°, which is 30 terms.We need to compute sum_{k=1}^{30} cos(2x_k) = sum_{k=1}^{30} cos(6° + (k-1)*12°)As we saw earlier, this sum is zero because it's a geometric series with ratio e^(i12°), which sums to zero.Therefore, sum = 15 - (1/2)(0) = 15.So, the correct answer should be 15.But in the assistant's solution, they concluded 30. So, perhaps the assistant made a mistake in the calculation.Alternatively, maybe I misapplied the identity.Wait, let me check the identity again.sin²(x) = (1 - cos(2x))/2Yes, that's correct.So, sum sin²(x) = (1/2) sum (1 - cos(2x)) = (1/2)(n - sum cos(2x))Where n is the number of terms, which is 30.So, sum sin²(x) = 15 - (1/2) sum cos(2x)If sum cos(2x) = 0, then sum sin²(x) = 15.Therefore, the correct answer is 15.But in the assistant's solution, they wrote:The sum is equivalent to 30 - (1/2)(0) = 30.Which suggests they thought the sum was 30 - (1/2)(sum cos terms), but according to the identity, it's (1/2)(30 - sum cos terms).So, the assistant incorrectly multiplied 30 by 1 instead of 1/2.Therefore, the correct answer is 15.But wait, let me think again. Maybe the assistant considered the sum as 30 terms each contributing 1, so 30, and then subtracted (1/2) sum cos terms.But according to the identity, it's (1/2)(sum 1 - sum cos terms). So, (1/2)(30 - sum cos terms). Therefore, 15 - (1/2) sum cos terms.Since sum cos terms is zero, the sum is 15.Therefore, the correct answer is 15.But in the assistant's solution, they concluded 30, which is incorrect.Therefore, I think the correct answer is 15.Wait, but let me check with another approach to confirm.Another way to compute the sum of sin²(x) over an arithmetic sequence of angles.I recall that for equally spaced angles, the sum of sin² can be computed using the formula:sum_{k=0}^{n-1} sin²(a + kd) = (n/2) - (1/2) * [sin(n d) / sin(d)] * cos(a + (n - 1)d/2 )But I'm not sure about this formula. Maybe it's better to use the identity we used earlier.Alternatively, perhaps using the identity for the sum of sin² terms.Wait, another approach: note that sin²(x) = (1 - cos(2x))/2, so sum sin²(x) = (n/2) - (1/2) sum cos(2x)As we did before.Since we have 30 terms, n = 30.Therefore, sum sin²(x) = 15 - (1/2) sum cos(2x)We need to compute sum cos(2x) where 2x are 6°, 18°, 30°, ..., 354°, which is 30 terms.As we saw earlier, this sum is zero because it's a geometric series with ratio e^(i12°), which sums to zero.Therefore, sum sin²(x) = 15 - 0 = 15.Therefore, the correct answer is 15.So, the assistant's solution was incorrect because they forgot to multiply by 1/2 when transforming the sum.Therefore, the correct answer is 15.

💡First, I need to find the midpoint of the segment with endpoints (10, 3) and (-4, 7). To do this, I'll use the midpoint formula, which averages the x-coordinates and the y-coordinates of the two endpoints.Calculating the x-coordinate of the midpoint:[frac{10 + (-4)}{2} = frac{6}{2} = 3]Calculating the y-coordinate of the midpoint:[frac{3 + 7}{2} = frac{10}{2} = 5]So, the midpoint is (3, 5).Next, I need to double the x-coordinate of this midpoint:[2 times 3 = 6]Now, the modified midpoint is (6, 5).Finally, I'll find the sum of the coordinates of this modified midpoint:[6 + 5 = 11]