💡Okay, so I have this problem where I need to find the range of values for ω such that the function f(x) = sin(ωx + π/3) has exactly three extreme points and two zeros in the interval (0, π). The options are given as A, B, C, D, and I need to figure out which one is correct.First, let me recall what extreme points and zeros mean in the context of a sine function. Extreme points occur where the derivative is zero, which for sin(θ) happens at θ = π/2 + kπ, where k is an integer. Zeros occur where the function crosses the x-axis, which for sin(θ) happens at θ = kπ, where k is an integer.So, f(x) = sin(ωx + π/3). Let's denote θ = ωx + π/3. Then, f(x) = sin(θ). The extreme points occur when θ = π/2 + kπ, and zeros occur when θ = kπ.Given that x is in the interval (0, π), θ will range from θ_min = ω*0 + π/3 = π/3 to θ_max = ω*π + π/3.So, θ ranges from π/3 to ωπ + π/3.Now, we need exactly two zeros in this interval. That means the equation sin(θ) = 0 should have exactly two solutions in θ ∈ (π/3, ωπ + π/3). The solutions to sin(θ) = 0 are θ = kπ. So, in the interval (π/3, ωπ + π/3), there should be exactly two integer multiples of π.Similarly, for the extreme points, we need exactly three extreme points, which correspond to θ = π/2 + kπ. So, the equation cos(θ) = 0 should have exactly three solutions in θ ∈ (π/3, ωπ + π/3). The solutions to cos(θ) = 0 are θ = π/2 + kπ.Let me tackle the zeros first.We need exactly two zeros, so we need exactly two values of k such that kπ ∈ (π/3, ωπ + π/3). Let's find the possible k.The smallest k such that kπ > π/3 is k=1, since 1π = π > π/3.The next one is k=2, 2π.The next one is k=3, 3π.We need exactly two zeros, so 2π must be less than ωπ + π/3, but 3π must be greater than ωπ + π/3.So, 2π < ωπ + π/3 < 3π.Divide all parts by π:2 < ω + 1/3 < 3Subtract 1/3:2 - 1/3 < ω < 3 - 1/3Which is:5/3 < ω < 8/3So, ω must be greater than 5/3 and less than 8/3 for exactly two zeros.Now, let's consider the extreme points. We need exactly three extreme points, which correspond to θ = π/2 + kπ.So, the equation cos(θ) = 0 has solutions at θ = π/2 + kπ.We need exactly three such solutions in θ ∈ (π/3, ωπ + π/3).Let's find the possible k.The first solution after θ = π/3 is θ = π/2 (k=0), then θ = 3π/2 (k=1), then θ = 5π/2 (k=2), then θ = 7π/2 (k=3), etc.We need exactly three solutions, so 5π/2 must be less than ωπ + π/3, but 7π/2 must be greater than ωπ + π/3.So, 5π/2 < ωπ + π/3 < 7π/2Divide all parts by π:5/2 < ω + 1/3 < 7/2Subtract 1/3:5/2 - 1/3 < ω < 7/2 - 1/3Convert to common denominators:(15/6 - 2/6) < ω < (21/6 - 2/6)Which is:13/6 < ω < 19/6So, ω must be greater than 13/6 and less than 19/6 for exactly three extreme points.Now, we have two conditions:From zeros: 5/3 < ω < 8/3From extremes: 13/6 < ω < 19/6We need ω to satisfy both conditions. So, we need the intersection of these two intervals.First, let's convert all fractions to have the same denominator to compare them easily.5/3 = 10/68/3 = 16/613/6 is already in sixths.19/6 is already in sixths.So, the intervals are:From zeros: 10/6 < ω < 16/6From extremes: 13/6 < ω < 19/6The intersection of these intervals is 13/6 < ω < 16/6, since 16/6 is less than 19/6.But wait, 16/6 is 8/3, which is the upper bound from the zeros condition.So, the intersection is 13/6 < ω < 8/3.But let's check if the endpoints are included or not.From zeros: We had 2π < ωπ + π/3 < 3π, which translates to 5/3 < ω < 8/3.Similarly, from extremes: 5π/2 < ωπ + π/3 < 7π/2, which translates to 13/6 < ω < 19/6.So, at ω = 13/6, does the function have exactly three extremes and two zeros?At ω = 13/6, θ_max = (13/6)π + π/3 = (13/6 + 2/6)π = 15/6 π = 5π/2.So, θ ranges from π/3 to 5π/2.For zeros: The zeros occur at θ = π, 2π, 3π.But θ_max is 5π/2, which is less than 3π (which is 6π/2). So, the zeros are at π and 2π, which are both less than 5π/2. So, exactly two zeros.For extremes: The extremes occur at θ = π/2, 3π/2, 5π/2.But θ_max is 5π/2, so the extremes are at π/2, 3π/2, and 5π/2. However, at θ = 5π/2, x = (5π/2 - π/3)/ω = (15π/6 - 2π/6)/(13/6) = (13π/6)/(13/6) = π. But x is in (0, π), so x=π is not included. Therefore, the extreme at θ=5π/2 occurs at x=π, which is not in the open interval (0, π). So, does that mean we only have two extremes?Wait, that's a problem. So, at ω=13/6, the third extreme is at x=π, which is not included. Therefore, in the open interval (0, π), we only have two extremes: at θ=π/2 and 3π/2.But we need exactly three extremes. Therefore, ω=13/6 is not included.Similarly, at ω=8/3, θ_max = (8/3)π + π/3 = (8π + π)/3 = 9π/3 = 3π.So, θ ranges from π/3 to 3π.For zeros: The zeros are at θ=π and 2π, which are both within (π/3, 3π). So, exactly two zeros.For extremes: The extremes are at θ=π/2, 3π/2, 5π/2, and 7π/2. But θ_max=3π, so 5π/2 is 2.5π, which is less than 3π, and 7π/2 is 3.5π, which is greater than 3π. So, the extremes are at π/2, 3π/2, and 5π/2. However, 5π/2 corresponds to x=(5π/2 - π/3)/(8/3) = (15π/6 - 2π/6)/(8/3) = (13π/6)/(8/3) = (13π/6)*(3/8) = 13π/16, which is less than π. So, all three extremes are within (0, π). Therefore, at ω=8/3, we have exactly three extremes.But wait, when ω=8/3, θ_max=3π, and the extreme at θ=5π/2 is still within (0, π). So, we have three extremes.But earlier, at ω=13/6, the third extreme is at x=π, which is not included. So, we need ω to be greater than 13/6 to ensure that the third extreme is within (0, π).Similarly, when ω approaches 19/6, θ_max = (19/6)π + π/3 = (19π/6 + 2π/6) = 21π/6 = 3.5π.So, θ ranges from π/3 to 3.5π.For extremes: The extremes are at π/2, 3π/2, 5π/2, 7π/2.But θ_max=3.5π=7π/2, so the extreme at 7π/2 is at x=(7π/2 - π/3)/(19/6) = (21π/6 - 2π/6)/(19/6) = (19π/6)/(19/6) = π. Again, x=π is not included, so the extreme at 7π/2 is at x=π, which is not in (0, π). Therefore, in (0, π), we have extremes at π/2, 3π/2, and 5π/2, which are all within (0, π). So, exactly three extremes.But if ω is equal to 19/6, then θ_max=7π/2, and the extreme at 7π/2 is at x=π, which is not included. So, we still have exactly three extremes in (0, π).Wait, but earlier, when ω=8/3, θ_max=3π, and the extreme at 5π/2 is within (0, π). So, the upper limit for ω is when θ_max=3π, which is ω=8/3.But wait, let's check the extremes again.If ω=19/6, θ_max=7π/2, which is 3.5π. The extremes are at π/2, 3π/2, 5π/2, and 7π/2. But 7π/2 is at x=π, which is excluded. So, in (0, π), we have extremes at π/2, 3π/2, and 5π/2. So, exactly three extremes.But if ω is slightly more than 19/6, say ω=19/6 + ε, then θ_max=7π/2 + επ. The extreme at 7π/2 + π/2=4π would be at x=(4π - π/3)/ω, which is greater than π, so it's outside the interval. So, the number of extremes remains three.Wait, but the problem states that there are exactly three extreme points and two zeros in (0, π). So, as long as θ_max is greater than 5π/2 but less than or equal to 7π/2, we have three extremes.But when θ_max exceeds 7π/2, the next extreme at 9π/2 would be at x=(9π/2 - π/3)/ω, which is greater than π if ω is too large.Wait, no, because θ_max=ωπ + π/3. So, if ω increases, θ_max increases. So, as ω increases beyond 19/6, θ_max increases beyond 7π/2, but the next extreme is at 9π/2, which would correspond to x=(9π/2 - π/3)/ω.But we need to ensure that this x is greater than π, so that it doesn't add another extreme within (0, π).So, to have exactly three extremes, θ_max must be less than 9π/2, but since θ_max=ωπ + π/3, we have ωπ + π/3 < 9π/2.Divide by π:ω + 1/3 < 9/2So, ω < 9/2 - 1/3 = 27/6 - 2/6 = 25/6 ≈4.1667.But our earlier condition from zeros was ω <8/3≈2.6667.So, the stricter condition is ω <8/3.Wait, but earlier, from extremes, we had ω <19/6≈3.1667.But 8/3≈2.6667 is less than 19/6≈3.1667.So, the intersection is ω <8/3.But wait, let's go back.From zeros: 5/3 < ω <8/3From extremes: 13/6 < ω <19/6Intersection is 13/6 < ω <8/3Because 13/6≈2.1667 and 8/3≈2.6667.So, the range is 13/6 < ω <8/3.But let's check the endpoints.At ω=13/6, θ_max=5π/2.As before, the extremes are at π/2, 3π/2, and 5π/2.But 5π/2 corresponds to x=π, which is excluded. So, in (0, π), we have only two extremes: π/2 and 3π/2.Therefore, ω must be greater than 13/6 to have three extremes.Similarly, at ω=8/3, θ_max=3π.The extremes are at π/2, 3π/2, and 5π/2.5π/2 corresponds to x=(5π/2 - π/3)/(8/3)= (15π/6 - 2π/6)/(8/3)= (13π/6)/(8/3)=13π/16, which is less than π. So, all three extremes are within (0, π).Therefore, ω=8/3 is included.Similarly, at ω=19/6, θ_max=7π/2.The extremes are at π/2, 3π/2, 5π/2, and 7π/2.7π/2 corresponds to x=π, which is excluded. So, in (0, π), we have three extremes: π/2, 3π/2, and 5π/2.Therefore, ω=19/6 is also included.But wait, from zeros, ω must be less than8/3≈2.6667, but 19/6≈3.1667 is greater than8/3.So, there's a conflict.Wait, no, because from zeros, we have ω <8/3, but from extremes, we have ω <19/6.So, the stricter condition is ω <8/3.But when ω=19/6, which is greater than8/3, the zeros condition is violated.Wait, let's check.At ω=19/6, θ_max=7π/2.Zeros occur at θ=π, 2π, 3π, 4π, etc.But θ_max=7π/2≈3.5π.So, zeros in (π/3,7π/2) are at π, 2π, 3π.But 3π is less than7π/2≈3.5π.So, zeros at π, 2π, 3π.But 3π is within (π/3,7π/2).So, that's three zeros, which violates the condition of exactly two zeros.Therefore, ω cannot be as high as19/6, because it would introduce a third zero.So, the upper limit for ω is when θ_max is just less than3π, so that the zero at3π is excluded.Wait, but θ_max=ωπ + π/3.We need θ_max <3π to exclude the third zero at3π.So, ωπ + π/3 <3πDivide by π:ω +1/3 <3So, ω <3 -1/3=8/3≈2.6667.So, ω must be less than8/3 to ensure that the third zero at3π is excluded.Therefore, the upper limit is8/3.Similarly, the lower limit is13/6≈2.1667.So, the range is13/6 <ω <8/3.But let's check if ω=8/3 is included.At ω=8/3, θ_max=3π.Zeros are at π and2π, which are both less than3π.So, exactly two zeros.Extremes are at π/2,3π/2,5π/2.5π/2 is less than3π, so all three extremes are within (0, π).Therefore, ω=8/3 is included.Similarly, at ω=13/6, θ_max=5π/2.Zeros are at π and2π, which are both less than5π/2.So, exactly two zeros.Extremes are at π/2,3π/2,5π/2.But5π/2 corresponds to x=π, which is excluded.So, in (0, π), we have two extremes.Therefore, ω=13/6 is excluded.So, the range is13/6 <ω ≤8/3.But looking at the options:A. [5/3,13/6)B. [5/3,19/6)C. (13/6,8/3]D. (13/6,19/6]So, our conclusion is13/6 <ω ≤8/3, which is option C.But wait, let me double-check.From zeros:5/3 <ω <8/3From extremes:13/6 <ω <19/6Intersection is13/6 <ω <8/3But since at ω=8/3, we still have exactly two zeros and three extremes, so ω=8/3 is included.Similarly, at ω=13/6, we have exactly two zeros but only two extremes, so ω=13/6 is excluded.Therefore, the range is(13/6,8/3], which is option C.

💡Alright, so I have this problem about a convex polyhedron, and I need to find the value of 100P + 10T + V, where P is the number of square faces meeting at a vertex, T is the number of triangular faces meeting at a vertex, and V is the total number of vertices. Hmm, okay.First, let me recall Euler's formula for convex polyhedrons: V - E + F = 2, where V is vertices, E is edges, and F is faces. The problem says there are 40 faces, each being either a triangle or a square. So, F = 40. Let me denote the number of triangular faces as T_faces and the number of square faces as P_faces. So, T_faces + P_faces = 40.At each vertex, exactly one triangular face and three square faces meet. That means, for each vertex, T = 1 and P = 3. Wait, but in the problem statement, P is the number of square faces meeting at a vertex, and T is the number of triangular faces meeting at a vertex. So, P = 3 and T = 1 for each vertex. So, I think I can use that to find some relationships.Now, I need to find V, the total number of vertices. To do that, I probably need to find E, the number of edges, because I can use Euler's formula once I have V and E. So, how do I find E?Well, each triangular face has 3 edges, and each square face has 4 edges. But each edge is shared by two faces. So, the total number of edges E can be calculated as (3*T_faces + 4*P_faces)/2.Let me write that down:E = (3*T_faces + 4*P_faces)/2Since T_faces + P_faces = 40, I can express one in terms of the other. Let's say T_faces = 40 - P_faces. Then,E = (3*(40 - P_faces) + 4*P_faces)/2E = (120 - 3*P_faces + 4*P_faces)/2E = (120 + P_faces)/2So, E = (120 + P_faces)/2Okay, now I need another equation to relate these variables. Maybe I can use the information about the vertices. Each vertex has one triangular face and three square faces meeting. So, each vertex is part of 1 triangle and 3 squares.Let me think about how many face-vertex incidences there are. Each triangular face has 3 vertices, and each square face has 4 vertices. So, the total number of face-vertex incidences is 3*T_faces + 4*P_faces.But each vertex has 1 triangular face and 3 square faces, so each vertex is part of 1 + 3 = 4 faces. Therefore, the total number of face-vertex incidences is also 4*V.So, 3*T_faces + 4*P_faces = 4*VBut T_faces = 40 - P_faces, so substituting:3*(40 - P_faces) + 4*P_faces = 4*V120 - 3*P_faces + 4*P_faces = 4*V120 + P_faces = 4*VSo, 4*V = 120 + P_facesTherefore, V = (120 + P_faces)/4Hmm, interesting. So, V is (120 + P_faces)/4, and E is (120 + P_faces)/2.Wait, so E is exactly twice V? Because (120 + P_faces)/2 is twice (120 + P_faces)/4. So, E = 2*V.That's an interesting relationship. So, E = 2V.Now, let's recall Euler's formula: V - E + F = 2We have F = 40, E = 2V, so substituting:V - 2V + 40 = 2- V + 40 = 2- V = 2 - 40- V = -38V = 38Wait, so V = 38? That seems a bit low, but let's check.If V = 38, then E = 2*38 = 76So, V - E + F = 38 - 76 + 40 = 2, which matches Euler's formula. Okay, so that seems consistent.Now, let's find P_faces. From earlier, we had:4*V = 120 + P_faces4*38 = 120 + P_faces152 = 120 + P_facesP_faces = 152 - 120P_faces = 32So, there are 32 square faces and T_faces = 40 - 32 = 8 triangular faces.So, T_faces = 8, P_faces = 32.Now, let's confirm the number of edges:E = (3*T_faces + 4*P_faces)/2E = (3*8 + 4*32)/2E = (24 + 128)/2E = 152/2E = 76Which matches our earlier calculation. Good.Now, let's think about the number of vertices. Each triangular face has 3 vertices, so total face-vertex incidences from triangles: 8*3 = 24Each square face has 4 vertices, so total face-vertex incidences from squares: 32*4 = 128Total face-vertex incidences: 24 + 128 = 152Since each vertex has 4 face incidences (1 triangle and 3 squares), the number of vertices V = 152 / 4 = 38, which matches our earlier result.Okay, so everything checks out so far.Now, the question is asking for 100P + 10T + V, where P is the number of square faces meeting at a vertex, and T is the number of triangular faces meeting at a vertex.From the problem statement: "At each of its V vertices, exactly one triangular face and three square faces meet." So, at each vertex, T = 1 and P = 3.Therefore, P = 3, T = 1, and V = 38.So, plugging into 100P + 10T + V:100*3 + 10*1 + 38 = 300 + 10 + 38 = 348Wait, but earlier, when I thought about the calculation, I got 351. Hmm, maybe I made a mistake somewhere.Wait, let me double-check. The problem says "the value of 100P + 10T + V where P is the number of square faces meeting at a vertex and T is the number of triangular faces meeting at a vertex."So, P = 3, T = 1, V = 38.So, 100*3 = 300, 10*1 = 10, and V = 38. So, 300 + 10 + 38 = 348.But in my initial thought process, I thought of 351. Maybe I confused P and T somewhere.Wait, let me go back. In the initial problem statement, it says "At each of its V vertices, exactly one triangular face and three square faces meet." So, T = 1, P = 3.So, P = 3, T = 1.Therefore, 100P + 10T + V = 100*3 + 10*1 + 38 = 300 + 10 + 38 = 348.But in my initial calculation, I thought of 351. Maybe I made a mistake there.Wait, let me check the initial calculation again.In the initial thought process, I had:E = (160 - T)/2, and then V = (160 - T)/2 - 38.But wait, that seems inconsistent with the correct approach.Wait, in the correct approach, we have:From face-vertex incidences: 3*T_faces + 4*P_faces = 4*VAnd T_faces + P_faces = 40So, we can solve for T_faces and P_faces.From 3*T_faces + 4*P_faces = 4*VAnd T_faces + P_faces = 40Let me solve these equations.Let me denote T_faces = t, P_faces = p.So,t + p = 403t + 4p = 4VFrom the first equation, t = 40 - pSubstitute into the second equation:3*(40 - p) + 4p = 4V120 - 3p + 4p = 4V120 + p = 4VSo, V = (120 + p)/4Also, E = (3t + 4p)/2 = (3*(40 - p) + 4p)/2 = (120 - 3p + 4p)/2 = (120 + p)/2So, E = (120 + p)/2From Euler's formula: V - E + F = 2V - E + 40 = 2V - E = -38But V = (120 + p)/4, E = (120 + p)/2So,(120 + p)/4 - (120 + p)/2 = -38Multiply both sides by 4 to eliminate denominators:(120 + p) - 2*(120 + p) = -152120 + p - 240 - 2p = -152-120 - p = -152So, -p = -152 + 120-p = -32p = 32So, P_faces = 32, T_faces = 8Then, V = (120 + 32)/4 = 152/4 = 38E = (120 + 32)/2 = 152/2 = 76So, everything checks out.Therefore, P = 3, T = 1, V = 38So, 100P + 10T + V = 100*3 + 10*1 + 38 = 300 + 10 + 38 = 348Wait, but in the initial thought process, I thought of 351. Maybe I confused P and T or made a calculation error.Wait, let me check the initial thought process again.In the initial thought process, I had:E = (160 - T)/2But where did 160 come from? Wait, in the initial thought process, I had:E = (3T + 4P)/2, and since T + P = 40, I substituted P = 40 - T, so E = (3T + 4*(40 - T))/2 = (3T + 160 - 4T)/2 = (160 - T)/2But in reality, T_faces is 8, so E = (3*8 + 4*32)/2 = (24 + 128)/2 = 152/2 = 76So, E = 76Then, V = E - 38 = 76 - 38 = 38Wait, but in the initial thought process, I had:V = E - 38But Euler's formula is V - E + F = 2, so V = E - F + 2 = E - 40 + 2 = E - 38So, that's correct.Then, substituting E = (160 - T)/2, and T = 1, so E = (160 - 1)/2 = 159/2 = 79.5Wait, that can't be, because E must be an integer.Wait, so in the initial thought process, I incorrectly assumed T = 1, but T is the number of triangular faces meeting at a vertex, which is 1, but T_faces is 8.So, I think I confused T_faces with T in the initial thought process.Therefore, the correct approach is to use T_faces = 8, P_faces = 32, V = 38So, 100P + 10T + V = 100*3 + 10*1 + 38 = 348Therefore, the correct answer is 348.But wait, in the initial thought process, I got 351, which was incorrect because I confused T_faces with T.So, the correct answer is 348.But let me double-check everything again to be sure.Given:- 40 faces, all triangles or squares.- At each vertex, 1 triangle and 3 squares meet.So, each vertex has degree 4 (1 triangle + 3 squares).Total face-vertex incidences: 3*T_faces + 4*P_faces = 4*VAlso, T_faces + P_faces = 40So, solving:From T_faces + P_faces = 40, T_faces = 40 - P_facesSubstitute into 3*T_faces + 4*P_faces = 4*V:3*(40 - P_faces) + 4*P_faces = 4*V120 - 3*P_faces + 4*P_faces = 4*V120 + P_faces = 4*VSo, V = (120 + P_faces)/4Also, E = (3*T_faces + 4*P_faces)/2 = (3*(40 - P_faces) + 4*P_faces)/2 = (120 - 3*P_faces + 4*P_faces)/2 = (120 + P_faces)/2From Euler's formula: V - E + F = 2V - E + 40 = 2V - E = -38But V = (120 + P_faces)/4, E = (120 + P_faces)/2So,(120 + P_faces)/4 - (120 + P_faces)/2 = -38Multiply both sides by 4:(120 + P_faces) - 2*(120 + P_faces) = -152120 + P_faces - 240 - 2*P_faces = -152-120 - P_faces = -152So, -P_faces = -32P_faces = 32Thus, T_faces = 8Then, V = (120 + 32)/4 = 152/4 = 38E = (120 + 32)/2 = 152/2 = 76So, V = 38, E = 76, F = 40Euler's formula: 38 - 76 + 40 = 2, which is correct.Now, P = 3 (number of square faces meeting at a vertex), T = 1 (number of triangular faces meeting at a vertex), V = 38So, 100P + 10T + V = 100*3 + 10*1 + 38 = 300 + 10 + 38 = 348Yes, that seems correct.I think in the initial thought process, I confused T_faces with T, which led to an incorrect calculation. So, the correct answer is 348.

💡First, I need to determine the total amount Jason spent on the flute and the song book.I'll add the cost of the flute, which is 142.46, to the cost of the song book, which is 7.00.This gives me a subtotal of 149.46.Next, I'll subtract this subtotal from the total amount Jason spent at the music store, which is 158.35.By subtracting 149.46 from 158.35, I can find out how much Jason spent on the music tool.The result of this subtraction is 8.89.Therefore, the music tool cost 8.89.

💡Okay, so I have this matrix that I need to raise to the 500th power. The matrix is:[begin{pmatrix} 1 & 0 -1 & 1 end{pmatrix}]Hmm, that's a 2x2 matrix. I remember that when dealing with matrices, especially when raising them to a power, it's helpful to look for patterns or properties that can simplify the process. Maybe I can find a pattern by computing the first few powers of the matrix and see if there's a trend.Let me start by computing the square of the matrix. So, let's compute:[begin{pmatrix} 1 & 0 -1 & 1 end{pmatrix} times begin{pmatrix} 1 & 0 -1 & 1 end{pmatrix}]To multiply these matrices, I'll use the standard matrix multiplication method. The element in the first row, first column is (1*1 + 0*(-1)) = 1 + 0 = 1. The element in the first row, second column is (1*0 + 0*1) = 0 + 0 = 0. The element in the second row, first column is (-1*1 + 1*(-1)) = -1 -1 = -2. The element in the second row, second column is (-1*0 + 1*1) = 0 + 1 = 1. So, the square of the matrix is:[begin{pmatrix} 1 & 0 -2 & 1 end{pmatrix}]Interesting, the (2,1) entry is now -2. Let me compute the cube to see if there's a pattern.Cube of the matrix:[begin{pmatrix} 1 & 0 -2 & 1 end{pmatrix} times begin{pmatrix} 1 & 0 -1 & 1 end{pmatrix}]Again, multiplying these:First row, first column: 1*1 + 0*(-1) = 1 + 0 = 1First row, second column: 1*0 + 0*1 = 0 + 0 = 0Second row, first column: (-2)*1 + 1*(-1) = -2 -1 = -3Second row, second column: (-2)*0 + 1*1 = 0 + 1 = 1So, the cube is:[begin{pmatrix} 1 & 0 -3 & 1 end{pmatrix}]Hmm, the (2,1) entry is now -3. It seems like each time I multiply by the original matrix, the (2,1) entry decreases by 1. So, the first power had -1, the square had -2, the cube had -3. If this pattern continues, then the nth power of the matrix would have -n in the (2,1) position.Let me test this hypothesis by computing the fourth power.Fourth power:[begin{pmatrix} 1 & 0 -3 & 1 end{pmatrix} times begin{pmatrix} 1 & 0 -1 & 1 end{pmatrix}]Multiplying:First row, first column: 1*1 + 0*(-1) = 1First row, second column: 1*0 + 0*1 = 0Second row, first column: (-3)*1 + 1*(-1) = -3 -1 = -4Second row, second column: (-3)*0 + 1*1 = 1So, the fourth power is:[begin{pmatrix} 1 & 0 -4 & 1 end{pmatrix}]Yep, the pattern holds. Each multiplication by the original matrix adds -1 to the (2,1) entry. So, it seems that for each exponent n, the (2,1) entry is -n.Therefore, if I want the 500th power, the (2,1) entry should be -500. The rest of the entries remain the same as the original matrix, which are 1s on the diagonal and 0 elsewhere.So, putting it all together, the 500th power of the matrix is:[begin{pmatrix} 1 & 0 -500 & 1 end{pmatrix}]I think that makes sense. To double-check, let me consider the general form of such matrices. The original matrix is a shear matrix, specifically a lower triangular matrix with 1s on the diagonal. Shear matrices have the property that when you multiply them, the off-diagonal entries add up. In this case, since the off-diagonal entry is -1, each multiplication adds another -1, hence after 500 multiplications, it's -500.Another way to think about it is using the concept of matrix exponentiation for shear matrices. For a matrix of the form:[begin{pmatrix} 1 & 0 a & 1 end{pmatrix}]Raising it to the nth power results in:[begin{pmatrix} 1 & 0 na & 1 end{pmatrix}]Since in our case, a = -1, so n*a = -n. Therefore, for n = 500, it's -500.This aligns with the pattern I observed earlier. So, I'm confident that the 500th power of the given matrix is indeed:[begin{pmatrix} 1 & 0 -500 & 1 end{pmatrix}]**Final Answer**The 500th power of the matrix is boxed{begin{pmatrix} 1 & 0 -500 & 1 end{pmatrix}}.

💡First, I need to determine the annual rent by multiplying the monthly rent by 12.Next, I'll calculate the total area of the shop by dividing the annual rent by the annual rent per square foot.Finally, I'll find the width of the shop by dividing the total area by the given length.

💡Okay, so I have this expression to evaluate: the square root of (9^6 + 3^12) divided by (9^3 + 3^17). Hmm, that looks a bit complicated, but maybe I can simplify it somehow. Let me think.First, I notice that both 9 and 3 are powers of 3. Specifically, 9 is 3 squared. So maybe I can rewrite everything in terms of 3 to make it easier. Let me try that.So, 9^6 is the same as (3^2)^6. When you raise a power to a power, you multiply the exponents. So that would be 3^(2*6) = 3^12. Okay, cool. Similarly, 9^3 is (3^2)^3, which is 3^(2*3) = 3^6.So now, the numerator of the expression inside the square root is 9^6 + 3^12, which I can rewrite as 3^12 + 3^12. That's just two times 3^12, right? So that simplifies to 2 * 3^12.Now, the denominator is 9^3 + 3^17. I already know that 9^3 is 3^6, so that part is easy. The other term is 3^17, which is already in terms of 3. So the denominator is 3^6 + 3^17.Hmm, I wonder if I can factor something out from the denominator to simplify it further. Let's see. Both terms have a 3^6 in them, right? So I can factor out 3^6 from both terms. That would give me 3^6 * (1 + 3^(17-6)) because when you factor out an exponent, you subtract the exponents. So that would be 3^6 * (1 + 3^11).So now, the entire expression inside the square root is (2 * 3^12) divided by (3^6 * (1 + 3^11)). Let me write that out: (2 * 3^12) / (3^6 * (1 + 3^11)).I can simplify this further by subtracting exponents when dividing. So 3^12 divided by 3^6 is 3^(12-6) = 3^6. So now the expression becomes (2 * 3^6) / (1 + 3^11).Okay, so now I have 2 * 3^6 divided by (1 + 3^11). Let me calculate the numerical values to see if that helps. 3^6 is 729, so 2 * 729 is 1458. And 3^11 is 177147. So the denominator is 1 + 177147, which is 177148.So now, the expression inside the square root is 1458 divided by 177148. Let me compute that division. 1458 divided by 177148. Hmm, that's a small number. Maybe I can approximate it.Let me see, 177148 divided by 1458 is roughly... Let me do 1458 times 100 is 145800, which is close to 177148. So 1458 * 121 is approximately 177148 because 1458 * 100 is 145800, and 1458 * 20 is 29160, so 145800 + 29160 = 174960, and then 1458 * 1 is 1458, so 174960 + 1458 = 176418. That's still less than 177148. So maybe 121.5? Let me check 1458 * 121.5. 1458 * 120 is 174960, and 1458 * 1.5 is 2187, so total is 174960 + 2187 = 177147. Oh, that's very close to 177148. So 1458 * 121.5 ≈ 177147, which is almost 177148. So 1458 / 177148 is approximately 1 / 121.5, which is roughly 0.00823.So now, the expression inside the square root is approximately 0.00823. Taking the square root of that would give me the square root of 0.00823. Let me think about what that is.I know that the square of 0.09 is 0.0081, because 0.09 * 0.09 = 0.0081. And 0.091 squared is 0.091 * 0.091. Let me calculate that: 0.09 * 0.09 is 0.0081, and then 0.001 * 0.09 is 0.00009, and 0.09 * 0.001 is another 0.00009, and 0.001 * 0.001 is 0.000001. So adding those up: 0.0081 + 0.00009 + 0.00009 + 0.000001 = 0.008281. Hmm, that's a bit higher than 0.00823.So 0.091 squared is approximately 0.008281, which is slightly higher than 0.00823. So maybe the square root of 0.00823 is a bit less than 0.091. Maybe around 0.0908 or something like that. But looking at the options given, 0.091 is one of them, so that must be the closest.Alternatively, maybe I made a miscalculation somewhere. Let me double-check my steps.Starting from the beginning: 9^6 is 3^12, and 3^12 is also 3^12, so numerator is 2 * 3^12. Denominator is 9^3 + 3^17, which is 3^6 + 3^17. Factoring out 3^6 gives 3^6(1 + 3^11). So the expression becomes (2 * 3^12) / (3^6(1 + 3^11)) = (2 * 3^6) / (1 + 3^11). That's correct.Then, 3^6 is 729, so 2 * 729 is 1458. 3^11 is 177147, so denominator is 1 + 177147 = 177148. So 1458 / 177148 ≈ 0.00823. Square root of that is approximately 0.0908, which is roughly 0.091.So I think that's correct. Therefore, the answer should be 0.091, which is option B.