💡Okay, so I need to figure out how much profit John makes from selling these woodburnings. Let me start by understanding the problem step by step.First, John sells 20 woodburnings, and each one is sold for 15. So, to find out how much money he makes in total from selling them, I should multiply the number of woodburnings by the price per piece. That would be 20 times 15. Let me write that down: 20 * 15. Hmm, 20 times 15 is 300. So, John makes 300 from selling the woodburnings.Now, I need to consider his costs. The problem says that the wood cost him 100. That means he spent 100 to get the materials needed to create these woodburnings. To find out how much profit he made, I need to subtract his total costs from his total revenue.So, his total revenue is 300, and his total cost is 100. Subtracting the cost from the revenue gives me the profit. That would be 300 minus 100, which equals 200. Therefore, John makes a profit of 200.Wait, let me double-check that to make sure I didn't make any mistakes. He sells 20 items at 15 each, so 20 * 15 is indeed 300. His cost is 100, so 300 minus 100 is 200. Yeah, that seems right. So, John's profit is 200.

💡Okay, so I have this quadratic function f(x) = ax² + bx + c. There are two parts to the problem. Let me tackle them one by one.Starting with part (1). The function satisfies f(-1 - x) = f(-1 + x) for any x. Hmm, that sounds like it's symmetric about the line x = -1. So, the axis of symmetry is x = -1. For a quadratic function, the axis of symmetry is given by x = -b/(2a). So, setting that equal to -1, I get:-b/(2a) = -1Multiplying both sides by 2a gives:-b = -2aSo, b = 2a.Alright, that's one equation relating b and a.Also, they told me that f(0) = 1. Plugging x = 0 into f(x):f(0) = a*(0)² + b*(0) + c = c = 1So, c = 1.Another piece of information is that the minimum value of f(x) is 0. Since it's a quadratic function, it has a minimum (since the coefficient a is positive, I assume; but wait, actually, they didn't specify if it's opening upwards or downwards. But since the minimum is 0, it must open upwards, so a is positive.The minimum value of a quadratic function f(x) = ax² + bx + c is given by f(-b/(2a)) = c - b²/(4a). So, setting that equal to 0:c - b²/(4a) = 0We already know c = 1, so:1 - b²/(4a) = 0Which simplifies to:b²/(4a) = 1So, b² = 4aBut from earlier, we have b = 2a. Let's substitute that into this equation:(2a)² = 4aSo, 4a² = 4aDivide both sides by 4:a² = aSubtract a from both sides:a² - a = 0Factor:a(a - 1) = 0So, a = 0 or a = 1But a can't be 0 because then it wouldn't be a quadratic function. So, a = 1.Then, since b = 2a, b = 2*1 = 2.So, putting it all together, f(x) = ax² + bx + c = 1x² + 2x + 1.Wait, that simplifies to f(x) = (x + 1)². Let me check if that makes sense.The axis of symmetry is x = -1, which matches the given condition f(-1 - x) = f(-1 + x). The vertex is at (-1, 0), which is the minimum point, and f(0) = (0 + 1)² = 1, which also matches. So, that seems correct.Moving on to part (2). Here, a = 1 and c = 0, so f(x) = x² + bx. We need |f(x)| ≤ 1 for all x in (0,1]. So, the function must stay between -1 and 1 on that interval.First, let's write down the inequality:-1 ≤ x² + bx ≤ 1 for all x ∈ (0,1]So, this is equivalent to two inequalities:1. x² + bx ≤ 12. x² + bx ≥ -1We need both of these to hold for all x in (0,1].Let me handle each inequality separately.Starting with the first inequality: x² + bx ≤ 1We can rearrange this as:bx ≤ 1 - x²Since x is in (0,1], x is positive. So, we can divide both sides by x without changing the inequality direction:b ≤ (1 - x²)/xSimplify the right-hand side:(1 - x²)/x = 1/x - xSo, b ≤ 1/x - x for all x ∈ (0,1]Therefore, b must be less than or equal to the minimum value of 1/x - x on (0,1].Similarly, for the second inequality: x² + bx ≥ -1Rearranging:bx ≥ -1 - x²Again, x is positive, so we can divide both sides by x:b ≥ (-1 - x²)/x = -1/x - xSo, b must be greater than or equal to -1/x - x for all x ∈ (0,1]Therefore, b must be greater than or equal to the maximum value of -1/x - x on (0,1].So, now, I need to find the minimum of 1/x - x on (0,1] and the maximum of -1/x - x on (0,1].Let's first find the minimum of 1/x - x on (0,1].Let me define g(x) = 1/x - x.To find its minimum, I can take the derivative and set it to zero.g'(x) = -1/x² - 1Set g'(x) = 0:-1/x² - 1 = 0-1/x² = 1Multiply both sides by x²:-1 = x²But x² is always positive, so this equation has no real solutions. Hmm, that means g(x) has no critical points in (0,1]. So, the extrema must occur at the endpoints.So, let's evaluate g(x) as x approaches 0+ and at x = 1.As x approaches 0+, 1/x approaches infinity, so g(x) approaches infinity.At x = 1, g(1) = 1/1 - 1 = 0.Therefore, on (0,1], the function g(x) = 1/x - x decreases from infinity to 0. So, the minimum value is 0, achieved at x = 1.Therefore, for the first inequality, b must be ≤ 0.Now, moving on to the second inequality: b ≥ -1/x - x for all x ∈ (0,1].Define h(x) = -1/x - x.We need to find the maximum of h(x) on (0,1].Again, let's take the derivative:h'(x) = 1/x² - 1Set h'(x) = 0:1/x² - 1 = 01/x² = 1x² = 1x = ±1But since x ∈ (0,1], the critical point is at x = 1.Evaluate h(x) at x = 1:h(1) = -1/1 - 1 = -2Also, as x approaches 0+, h(x) = -1/x - x approaches negative infinity.So, the function h(x) increases from negative infinity to -2 as x goes from 0 to 1.Therefore, the maximum value of h(x) on (0,1] is -2, achieved at x = 1.Thus, for the second inequality, b must be ≥ -2.Putting both inequalities together:-2 ≤ b ≤ 0Therefore, the range of b is [-2, 0].Let me double-check this result.If b = 0, then f(x) = x², which on (0,1] is between 0 and 1, so |f(x)| ≤ 1 holds.If b = -2, then f(x) = x² - 2x. Let's check the maximum and minimum on (0,1].f(x) = x² - 2x = x(x - 2). On (0,1], the vertex is at x = 1, f(1) = 1 - 2 = -1. At x approaching 0, f(x) approaches 0. So, the function goes from 0 down to -1 at x=1, so |f(x)| ≤ 1 holds.What about somewhere in between, say b = -1. Then f(x) = x² - x. The vertex is at x = 0.5, f(0.5) = 0.25 - 0.5 = -0.25. So, the minimum is -0.25, which is within the bounds. At x=1, f(1) = 1 - 1 = 0. So, |f(x)| ≤ 1 holds.If b were greater than 0, say b = 1, then f(x) = x² + x. At x=1, f(1) = 1 + 1 = 2, which exceeds 1, so |f(x)| would be 2, violating the condition. Similarly, if b were less than -2, say b = -3, then f(x) = x² - 3x. At x=1, f(1) = 1 - 3 = -2, which is outside the |f(x)| ≤ 1 condition. So, indeed, b must be between -2 and 0.Therefore, the range of b is [-2, 0].**Final Answer**(1) The analytic expression of ( f(x) ) is (boxed{(x + 1)^2}).(2) The range of ( b ) is (boxed{[-2, 0]}).

💡Okay, so I have this problem here: If the determinant of matrix M is 3, I need to find the determinant of M cubed, which is det(M³). Hmm, determinants... I remember they have something to do with the scaling factor of the linear transformation represented by the matrix, but I'm not entirely sure how that applies here. Let me think.First, I recall that determinants have some properties, especially when it comes to matrix multiplication and powers. I think one of the properties is that the determinant of a product of matrices is the product of their determinants. So, det(AB) = det(A) * det(B). If that's the case, then maybe det(M³) is just det(M) multiplied by itself three times?Wait, let me make sure. So, M³ means M multiplied by M multiplied by M, right? So, det(M³) would be det(M * M * M). Using the property I just remembered, that should be det(M) * det(M) * det(M). Which is det(M) cubed. So, if det(M) is 3, then det(M³) should be 3³, which is 27. That seems straightforward.But let me double-check if there are any exceptions or special cases I need to consider. For example, does this property hold for any square matrix, regardless of its size or whether it's invertible? I think determinants are defined for square matrices, and as long as M is a square matrix, this property should hold. Since the problem didn't specify the size of M, I guess it's safe to assume it's a square matrix.Another thing I'm wondering about is whether M needs to be invertible for this property to work. Well, if M is invertible, its determinant is non-zero, which is true here since det(M) = 3, which is not zero. So, M is invertible, and that should be fine. But even if M wasn't invertible, as long as it's square, the determinant properties should still apply, right? So, I don't think invertibility affects this particular property.Let me also recall if there are any other properties related to determinants and powers. I think there's a general formula: det(Mⁿ) = (det(M))ⁿ for any integer n. So, in this case, n is 3, so it's (det(M))³. That aligns with what I thought earlier. So, applying that formula, det(M³) = (det(M))³ = 3³ = 27.Just to make sure I'm not missing anything, let me think about a specific example. Suppose M is a 2x2 matrix with determinant 3. Let's say M = [[a, b], [c, d]] and ad - bc = 3. Then, M² would be [[a² + bc, ab + bd], [ac + cd, bc + d²]], and det(M²) would be (a² + bc)(bc + d²) - (ab + bd)(ac + cd). That seems complicated, but if I compute it, it should equal (det(M))² = 9. Similarly, M³ would be M*M*M, and det(M³) should be 27. So, even in a concrete example, the determinant of the cube should be the cube of the determinant.Wait, but calculating M² and M³ manually might be tedious. Maybe I can use diagonal matrices as a simpler example. If M is a diagonal matrix with entries λ₁ and λ₂ on the diagonal, then det(M) = λ₁λ₂. If I cube M, the diagonal entries become λ₁³ and λ₂³, so det(M³) = λ₁³λ₂³ = (λ₁λ₂)³ = (det(M))³. So, that works out too.Another thought: eigenvalues. The determinant of a matrix is the product of its eigenvalues. If I raise the matrix to the power of 3, the eigenvalues get raised to the power of 3 as well. So, the determinant, being the product of the eigenvalues, would also be raised to the power of 3. That makes sense.Is there any case where this wouldn't hold? Well, if M is not a square matrix, then determinants aren't defined, so that's a given. But since the problem mentions det(M), M must be square. Also, if M is not diagonalizable, does that affect the determinant of its power? I don't think so because the determinant depends on the eigenvalues, regardless of whether the matrix is diagonalizable or not. So, even if M is defective (not diagonalizable), det(M³) should still be (det(M))³.Let me also think about the properties of determinants in terms of matrix similarity. If two matrices are similar, they have the same determinant. If M is similar to another matrix N, then M = P⁻¹NP for some invertible matrix P. Then, M³ = P⁻¹N³P, and det(M³) = det(N³). But det(N³) = (det(N))³, and since det(M) = det(N), this again gives det(M³) = (det(M))³.So, all these different angles—direct computation, diagonal matrices, eigenvalues, similarity transformations—lead me to the same conclusion. Therefore, I feel confident that det(M³) is indeed 27 when det(M) is 3.Just to wrap up, the key steps are:1. Recognize that det(Mⁿ) = (det(M))ⁿ for any integer n.2. Apply this property with n = 3.3. Calculate (det(M))³ = 3³ = 27.I don't see any flaws in this reasoning, so I think the answer is correct.

💡First, I need to determine how many dance lessons Tom paid for. Each lesson costs 10, and he paid a total of 80. By dividing the total amount paid by the cost per lesson, I can find out how many lessons he paid for.Next, I should account for the free lessons. Tom receives two free lessons for every set of lessons he pays for. This means that for every paid lesson, he gets an additional free lesson.Finally, I will add the number of paid lessons to the number of free lessons to find the total number of dance lessons Tom took.

💡Okay, so I have this problem here: If x and y are integers, what is the least positive number of 24x + some integer times y, given that the least positive number for this integer is 4? Hmm, that sounds a bit confusing at first, but let me try to break it down.First, I think the problem is asking about the smallest positive number that can be expressed as 24x plus another integer multiplied by y, and it's given that the smallest positive number for that "some integer" is 4. Wait, maybe I'm misinterpreting that. Let me read it again."If x and y are integers, what is the least positive number of 24x + some integer times y, given that the least positive number for this integer is 4?"Okay, so maybe it's saying that the smallest positive number that can be expressed as 24x plus some integer times y is 4. So, we're looking for the smallest positive value of the expression 24x + ky, where k is some integer, and the smallest positive value this expression can take is 4.Alright, so that means 4 is the greatest common divisor (GCD) of 24 and k, right? Because the smallest positive linear combination of two integers is their GCD. So, if 4 is the smallest positive number that can be expressed as 24x + ky, then GCD(24, k) must be 4.So, to find k, we need to find an integer such that GCD(24, k) = 4. That means k must be a multiple of 4, but not a multiple of any larger common divisor of 24. Since 24 is 2^3 * 3, its divisors are 1, 2, 3, 4, 6, 8, 12, and 24. So, to have GCD(24, k) = 4, k must be a multiple of 4 but not a multiple of 8 or 12 or 24, because that would increase the GCD beyond 4.Wait, no, actually, k can be any multiple of 4, but when you take the GCD with 24, it should be 4. So, k must be such that when you factor it, the highest power of 2 it shares with 24 is 2^2, which is 4. So, k can be 4, 12, 20, etc., but not 8, 16, 24, etc., because those would share higher powers of 2 with 24, making the GCD larger than 4.But the problem says "the least positive number for this integer is 4." So, does that mean k itself is 4? Or does it mean that the smallest positive value of the expression 24x + ky is 4? I think it's the latter. So, the smallest positive value of 24x + ky is 4, which implies that GCD(24, k) = 4.Therefore, k must be such that GCD(24, k) = 4. So, k can be any integer where the GCD with 24 is 4. The smallest positive such k would be 4 itself, because GCD(24, 4) = 4.Wait, but if k is 4, then the expression becomes 24x + 4y. The smallest positive number that can be expressed as 24x + 4y is indeed 4, because 4 is the GCD of 24 and 4. So, that makes sense.But the problem says "some integer times y," so k is that integer. So, the least positive number for this integer k is 4. So, k is 4.Therefore, the expression is 24x + 4y, and the least positive number is 4.Wait, but the problem is asking for "the least positive number of 24x + some integer times y," which is 4. So, I think the answer is 4.But let me double-check. If k is 4, then 24x + 4y can indeed produce 4 as the smallest positive number because GCD(24,4)=4. So, yes, that seems correct.Alternatively, if k were something else, like 8, then GCD(24,8)=8, which would make the smallest positive number 8, which is larger than 4. So, to have the smallest positive number as 4, k must be 4.Therefore, the least positive number is 4.

💡Okay, so I have this problem where I need to find the equivalent standard spherical coordinate for the point (4, 3π/4, 7π/4). The constraints are that ρ should be positive, θ should be between 0 and 2π, and φ should be between 0 and π. First, let me recall what spherical coordinates are. In spherical coordinates, a point in space is represented by three values: ρ, θ, and φ. Here, ρ is the distance from the origin to the point, θ is the angle in the xy-plane from the positive x-axis, and φ is the angle from the positive z-axis down to the point. Given the point (4, 3π/4, 7π/4), I can see that ρ is 4, which is already positive, so that's fine. θ is 3π/4, which is 135 degrees, and that's within the range of 0 to 2π, so that also seems okay. However, φ is 7π/4, which is 315 degrees. But wait, φ is supposed to be between 0 and π, which is 0 to 180 degrees. So 315 degrees is outside of that range. Hmm, so I need to adjust φ to be within 0 to π. How do I do that? I remember that in spherical coordinates, if φ is greater than π, we can reflect the point to the opposite side of the origin by subtracting φ from 2π and adjusting θ accordingly. Let me write that down. If φ is greater than π, then the new φ should be 2π - φ. But wait, actually, it's π - (φ - π) because φ is measured from the positive z-axis. So if φ is 7π/4, which is more than π, I can subtract π from it to get the angle below the xy-plane, and then take that angle as the new φ from the positive z-axis. Wait, let me think again. If φ is 7π/4, which is 315 degrees, that's actually pointing below the xy-plane. Since φ is supposed to be the angle from the positive z-axis, it can't be more than π. So to get the equivalent point, I need to find an angle φ' such that φ' = 2π - φ, but since φ is measured from the positive z-axis, it's actually φ' = π - (φ - π) = 2π - φ. Wait, no, that might not be correct. Let me visualize this. If φ is 7π/4, which is 315 degrees, that's equivalent to -45 degrees from the positive z-axis. But since angles are typically measured as positive in the counterclockwise direction from the positive z-axis, going beyond π would point the point into the lower hemisphere. So to bring φ within 0 to π, I need to find the supplementary angle. That is, φ' = 2π - φ. But wait, 2π - 7π/4 is 8π/4 - 7π/4 = π/4. So φ' would be π/4. But I also need to adjust θ because reflecting the point across the origin would change the direction in the xy-plane. So if I subtract φ from 2π, I also need to add π to θ to point in the opposite direction. Wait, let me confirm. If I have a point with φ > π, I can represent it as a point with φ' = 2π - φ and θ' = θ + π. But since θ is already 3π/4, adding π would give θ' = 3π/4 + π = 7π/4. But 7π/4 is still within 0 to 2π, so that's okay. But wait, if I do that, the new point would be (4, 7π/4, π/4). Is that correct? Let me check. Alternatively, I could think of it as reflecting the point through the origin. So the original point is at (4, 3π/4, 7π/4). Reflecting it through the origin would invert the direction, so ρ remains the same, θ increases by π, and φ becomes π - (φ - π) = 2π - φ. Wait, that seems consistent with what I did earlier. So φ' = 2π - 7π/4 = π/4, and θ' = 3π/4 + π = 7π/4. But let me verify this by converting both points to Cartesian coordinates and see if they are negatives of each other, which they should be if we've reflected through the origin. The original point is (4, 3π/4, 7π/4). Converting to Cartesian coordinates:x = ρ sinφ cosθ = 4 sin(7π/4) cos(3π/4)y = 4 sin(7π/4) sin(3π/4)z = 4 cos(7π/4)sin(7π/4) = -√2/2, cos(7π/4) = √2/2cos(3π/4) = -√2/2, sin(3π/4) = √2/2So x = 4*(-√2/2)*(-√2/2) = 4*(2/4) = 2y = 4*(-√2/2)*(√2/2) = 4*(-2/4) = -2z = 4*(√2/2) = 2√2So the original point in Cartesian is (2, -2, 2√2).Now, the reflected point should be (-2, 2, -2√2). Let's see if (4, 7π/4, π/4) gives that.x = 4 sin(π/4) cos(7π/4)y = 4 sin(π/4) sin(7π/4)z = 4 cos(π/4)sin(π/4) = √2/2, cos(π/4) = √2/2cos(7π/4) = √2/2, sin(7π/4) = -√2/2x = 4*(√2/2)*(√2/2) = 4*(2/4) = 2y = 4*(√2/2)*(-√2/2) = 4*(-2/4) = -2z = 4*(√2/2) = 2√2Wait, that's the same as the original point. That can't be right. I must have made a mistake in my reflection.Wait, no, because if I reflect through the origin, I should have ρ negative, but ρ is supposed to be positive. So instead, to represent the same point with ρ positive, I need to adjust θ and φ accordingly.Wait, perhaps I should not add π to θ, but instead, since φ is being adjusted, I need to find the equivalent point by keeping ρ positive and adjusting θ and φ.Let me think differently. Since φ is 7π/4, which is more than π, I can subtract π from φ to get the angle below the xy-plane, and then add π to θ to point in the opposite direction.So φ' = 7π/4 - π = 7π/4 - 4π/4 = 3π/4. But wait, that's still more than π/2, but less than π. Wait, no, 3π/4 is 135 degrees, which is within 0 to π. So φ' = 3π/4.But then θ' = θ + π = 3π/4 + π = 7π/4.So the new point would be (4, 7π/4, 3π/4). Let me check this.Converting (4, 7π/4, 3π/4) to Cartesian:x = 4 sin(3π/4) cos(7π/4)y = 4 sin(3π/4) sin(7π/4)z = 4 cos(3π/4)sin(3π/4) = √2/2, cos(3π/4) = -√2/2cos(7π/4) = √2/2, sin(7π/4) = -√2/2x = 4*(√2/2)*(√2/2) = 4*(2/4) = 2y = 4*(√2/2)*(-√2/2) = 4*(-2/4) = -2z = 4*(-√2/2) = -2√2Wait, so the Cartesian coordinates are (2, -2, -2√2). But the original point was (2, -2, 2√2). These are not negatives of each other. The z-coordinate is inverted, but x and y are the same. That doesn't seem right.Wait, maybe I need to adjust θ differently. Let me try another approach. Since φ is 7π/4, which is equivalent to -π/4 in terms of direction from the positive z-axis. So to get the angle within 0 to π, I can take φ' = π/4, and then adjust θ by adding π to point in the opposite direction.So φ' = π/4, θ' = 3π/4 + π = 7π/4.Let's convert (4, 7π/4, π/4) to Cartesian:x = 4 sin(π/4) cos(7π/4)y = 4 sin(π/4) sin(7π/4)z = 4 cos(π/4)sin(π/4) = √2/2, cos(π/4) = √2/2cos(7π/4) = √2/2, sin(7π/4) = -√2/2x = 4*(√2/2)*(√2/2) = 4*(2/4) = 2y = 4*(√2/2)*(-√2/2) = 4*(-2/4) = -2z = 4*(√2/2) = 2√2Wait, that's the same as the original point. So that can't be right. I must be making a mistake in how I'm adjusting θ and φ.Let me try a different method. Since φ is 7π/4, which is more than π, I can find the reference angle by subtracting π from φ, which gives 7π/4 - π = 3π/4. Then, to get the correct direction, I need to add π to θ, so θ' = 3π/4 + π = 7π/4.So the new point would be (4, 7π/4, 3π/4). Let's convert this to Cartesian:x = 4 sin(3π/4) cos(7π/4)y = 4 sin(3π/4) sin(7π/4)z = 4 cos(3π/4)sin(3π/4) = √2/2, cos(3π/4) = -√2/2cos(7π/4) = √2/2, sin(7π/4) = -√2/2x = 4*(√2/2)*(√2/2) = 4*(2/4) = 2y = 4*(√2/2)*(-√2/2) = 4*(-2/4) = -2z = 4*(-√2/2) = -2√2So the Cartesian coordinates are (2, -2, -2√2). Comparing this to the original point's Cartesian coordinates (2, -2, 2√2), the z-coordinate is inverted. So this point is the reflection of the original point across the xy-plane. But I wanted to reflect it through the origin, which would invert all coordinates. Wait, so maybe I need to adjust θ differently. Instead of adding π to θ, perhaps I should subtract π or adjust it in another way. Let me think.If I have a point with φ > π, it's in the lower hemisphere. To represent it in the upper hemisphere, I can take φ' = 2π - φ, but since φ is measured from the positive z-axis, it's actually φ' = π - (φ - π) = 2π - φ. Wait, that's the same as before. But when I did that, I ended up with the same x and y but inverted z. To get the reflection through the origin, I need to invert all coordinates, so x becomes -x, y becomes -y, z becomes -z. So, to achieve that, I can keep ρ positive, adjust θ by adding π, and adjust φ to be π - (φ - π) = 2π - φ. Wait, let's try that. So φ' = 2π - 7π/4 = π/4. θ' = 3π/4 + π = 7π/4. So the new point is (4, 7π/4, π/4). Converting this to Cartesian:x = 4 sin(π/4) cos(7π/4) = 4*(√2/2)*(√2/2) = 2y = 4 sin(π/4) sin(7π/4) = 4*(√2/2)*(-√2/2) = -2z = 4 cos(π/4) = 4*(√2/2) = 2√2So the Cartesian coordinates are (2, -2, 2√2), which is the same as the original point. That means I haven't changed the point at all. That's not what I want. I need to get the reflection through the origin, which would be (-2, 2, -2√2). Wait, so maybe I need to adjust θ differently. Instead of adding π to θ, perhaps I should subtract π. Let's try θ' = 3π/4 - π = -π/4. But θ is supposed to be between 0 and 2π, so -π/4 is equivalent to 7π/4. So that's the same as before. Hmm, this is confusing. Maybe I need to approach this differently. Let me recall that in spherical coordinates, if φ > π, we can represent the point by setting φ' = 2π - φ and θ' = θ + π. But since θ is already 3π/4, adding π gives θ' = 7π/4, which is within 0 to 2π. But when I did that earlier, the Cartesian coordinates didn't change, which suggests that this method isn't working. Maybe I'm missing something. Wait, perhaps I should consider that when φ > π, the point is in the lower hemisphere, and to represent it in the upper hemisphere, I need to adjust both θ and φ. Specifically, φ' = 2π - φ, and θ' = θ + π. Let me try that again. φ' = 2π - 7π/4 = π/4. θ' = 3π/4 + π = 7π/4. So the new point is (4, 7π/4, π/4). Converting to Cartesian:x = 4 sin(π/4) cos(7π/4) = 4*(√2/2)*(√2/2) = 2y = 4 sin(π/4) sin(7π/4) = 4*(√2/2)*(-√2/2) = -2z = 4 cos(π/4) = 4*(√2/2) = 2√2Same as before. So this method isn't changing the point. I must be misunderstanding how to adjust θ and φ when φ > π.Wait, maybe I should think about the direction of θ. If I add π to θ, I'm pointing in the opposite direction in the xy-plane, but since φ is also being adjusted, it's not just a simple reflection. Alternatively, perhaps I should consider that when φ > π, the point is below the xy-plane, and to represent it above, I need to adjust both θ and φ. Let me try another approach. Let's consider the original point (4, 3π/4, 7π/4). Since φ = 7π/4 is more than π, I can subtract 2π from φ to get it within the range of -π to 0, but that's not helpful since we need it between 0 and π. Alternatively, I can find the angle from the positive z-axis by subtracting φ from 2π, but that might not be the right approach. Wait, perhaps I should think of φ as the angle from the positive z-axis, so if it's more than π, it's pointing downward. To get the equivalent upward angle, I can take φ' = 2π - φ, but since φ is measured from the positive z-axis, it's actually φ' = π - (φ - π) = 2π - φ. Wait, that's the same as before. So φ' = 2π - 7π/4 = π/4. Then, to adjust θ, I need to add π to it to point in the opposite direction. So θ' = 3π/4 + π = 7π/4. But as we saw earlier, this doesn't change the Cartesian coordinates, which suggests that this method isn't working. Maybe I'm overcomplicating this. Let me try to visualize the point. The original point has φ = 7π/4, which is 315 degrees from the positive z-axis, pointing downward. To represent this in the upper hemisphere, I need to find an angle φ' such that φ' = π - (φ - π) = 2π - φ. Wait, that's the same as before. So φ' = π/4. Then, to get the correct direction in the xy-plane, I need to add π to θ, so θ' = 3π/4 + π = 7π/4. But when I convert (4, 7π/4, π/4) to Cartesian, I get the same point as the original. That can't be right. Wait, maybe I need to adjust θ differently. Instead of adding π, perhaps I should subtract π. Let's try θ' = 3π/4 - π = -π/4, which is equivalent to 7π/4. So that's the same as before. I'm stuck. Maybe I should try a different method altogether. Let me recall that in spherical coordinates, if φ > π, the point is below the xy-plane, and to represent it above, we can set φ' = 2π - φ and θ' = θ + π. So φ' = 2π - 7π/4 = π/4. θ' = 3π/4 + π = 7π/4. But converting (4, 7π/4, π/4) to Cartesian gives the same point as the original, which is not what I want. Wait, maybe I need to invert ρ. If I set ρ' = -ρ, then θ' = θ + π, and φ' = φ. But ρ is supposed to be positive, so that's not allowed. Alternatively, perhaps I need to keep ρ positive and adjust θ and φ accordingly. Wait, let me think about the relationship between φ and θ. If I have a point below the xy-plane, I can represent it above by keeping the same ρ, adjusting θ by π, and setting φ' = π - φ. Wait, that might work. Let's try that. φ' = π - (φ - π) = 2π - φ. So φ' = 2π - 7π/4 = π/4. θ' = θ + π = 3π/4 + π = 7π/4. But again, converting (4, 7π/4, π/4) to Cartesian gives the same point as the original. I'm clearly missing something here. Maybe I need to adjust θ differently. Let me try subtracting π instead of adding. θ' = 3π/4 - π = -π/4, which is equivalent to 7π/4. So that's the same as before. Wait, perhaps I should consider that when φ > π, the point is in the lower hemisphere, and to represent it in the upper hemisphere, I need to set φ' = 2π - φ and θ' = θ + π. But since θ is already 3π/4, adding π gives θ' = 7π/4. But converting (4, 7π/4, π/4) to Cartesian gives the same point as the original, which suggests that this method isn't changing the point. I think I need to step back and consider that maybe the original point is already in the correct hemisphere, but φ is just measured beyond π. So perhaps I don't need to adjust θ at all, just adjust φ. Wait, but φ is supposed to be between 0 and π. So if φ is 7π/4, which is more than π, I need to adjust it. Let me try this: φ' = 2π - 7π/4 = π/4. Then, since I've effectively reflected the point through the xy-plane, I need to adjust θ by adding π to point in the opposite direction. So θ' = 3π/4 + π = 7π/4. But as before, converting (4, 7π/4, π/4) to Cartesian gives the same point as the original, which is not what I want. Wait, maybe I'm misunderstanding the relationship between θ and φ. Let me recall that θ is the angle in the xy-plane, and φ is the angle from the positive z-axis. So if φ is more than π, the point is below the xy-plane, and to represent it above, I need to adjust both θ and φ. Perhaps I should think of it as follows: if φ > π, then the point is below the xy-plane, so to bring it above, I can set φ' = 2π - φ, and θ' = θ + π. So φ' = 2π - 7π/4 = π/4. θ' = 3π/4 + π = 7π/4. But converting (4, 7π/4, π/4) to Cartesian gives the same point as the original, which is not what I want. Wait, maybe I need to invert the direction of θ. Instead of adding π, perhaps I should subtract π. θ' = 3π/4 - π = -π/4, which is equivalent to 7π/4. But that's the same as before. I'm clearly stuck here. Maybe I should look for another approach. Let me try converting the original point to Cartesian coordinates and then back to spherical coordinates, ensuring that φ is within 0 to π. Original point: (4, 3π/4, 7π/4). Converting to Cartesian:x = 4 sin(7π/4) cos(3π/4)y = 4 sin(7π/4) sin(3π/4)z = 4 cos(7π/4)sin(7π/4) = -√2/2, cos(7π/4) = √2/2cos(3π/4) = -√2/2, sin(3π/4) = √2/2x = 4*(-√2/2)*(-√2/2) = 4*(2/4) = 2y = 4*(-√2/2)*(√2/2) = 4*(-2/4) = -2z = 4*(√2/2) = 2√2So Cartesian coordinates are (2, -2, 2√2).Now, to convert back to spherical coordinates with φ between 0 and π:ρ = √(x² + y² + z²) = √(4 + 4 + 8) = √16 = 4θ = arctan(y/x) = arctan(-2/2) = arctan(-1) = -π/4. But since x is positive and y is negative, θ is in the fourth quadrant, so θ = 7π/4.φ = arccos(z/ρ) = arccos(2√2/4) = arccos(√2/2) = π/4.So the equivalent spherical coordinates are (4, 7π/4, π/4).Wait, but that's different from what I thought earlier. So the correct equivalent point is (4, 7π/4, π/4). But earlier, when I tried to adjust θ and φ, I ended up with the same Cartesian coordinates as the original point, which confused me. But now, by converting to Cartesian and back, I see that the correct equivalent point is (4, 7π/4, π/4). So, to summarize, the original point (4, 3π/4, 7π/4) has φ = 7π/4, which is more than π. To bring φ within 0 to π, I need to adjust both φ and θ. By converting to Cartesian and back, I found that the equivalent point is (4, 7π/4, π/4). Therefore, the standard spherical coordinates for the given point are (4, 7π/4, π/4).