answer：Okay, so I have this problem about a sphere with radius R, and there are two points A and B on its surface. The angle between them, specifically the angle AOB, is π/6. I need to find the spherical distance between points A and B. Hmm, let me think about how to approach this.First, I remember that on a sphere, the distance between two points isn't just a straight line through the sphere, but rather the shortest path along the surface. This is called a great circle distance. A great circle is any circle drawn on the sphere that has the same center and radius as the sphere itself. So, the distance between A and B would be the length of the arc of the great circle connecting them.Now, the angle AOB is given as π/6. I think this is the central angle between points A and B, with O being the center of the sphere. So, if I can find the length of the arc corresponding to this central angle, that should give me the spherical distance.I recall that the length of an arc (s) in a circle is given by the formula s = rθ, where r is the radius of the circle and θ is the central angle in radians. In this case, the radius of the sphere is R, and the central angle is π/6. So, plugging these values into the formula, I should get s = R * (π/6).Wait, let me make sure I'm not missing anything here. Is there anything else I need to consider? For example, is the angle given in degrees or radians? The problem states it's π/6, which is in radians, so that's consistent with the formula I used. Also, since both points are on the surface of the sphere, the radius R is the same for both points from the center O.Is there a possibility that the angle given isn't the central angle? Well, the problem says angle AOB, which is the angle at the center of the sphere between points A and B. So, yes, that should be the central angle. Therefore, using the arc length formula should be appropriate here.Let me visualize this. Imagine the sphere with center O, and two points A and B on its surface. The angle between OA and OB is π/6. If I draw the great circle that passes through A and B, the arc from A to B along this circle is the shortest path on the sphere's surface, which is the spherical distance we're looking for.So, calculating this distance, it's just the arc length corresponding to the central angle π/6. Since the radius is R, the formula s = Rθ gives s = R*(π/6). That seems straightforward.I don't think I need to consider the chord length here because the question specifically asks for the spherical distance, which is along the surface, not through the interior of the sphere. The chord length would be a straight line between A and B, but that's not what's being asked.Let me double-check the formula. Arc length s = Rθ. Yes, that's correct. θ is in radians, which it is here (π/6). So, s = R*(π/6) is the right calculation.Is there any unit conversion needed? The angle is given in radians, so no conversion is necessary. If it were in degrees, I would have to convert it to radians first, but since it's already in radians, I can proceed as is.So, putting it all together, the spherical distance between points A and B is πR/6.Wait, just to make sure, let me think about another way to approach this. Maybe using the law of cosines for spherical triangles? But that might be overcomplicating things for this problem.In spherical trigonometry, the law of cosines is used to relate the sides and angles of a spherical triangle. However, in this case, since we're dealing with a central angle and the radius is given, the simpler arc length formula should suffice.Alternatively, if I consider the triangle AOB, which is a triangle with two sides equal to R (since OA and OB are radii of the sphere) and the included angle π/6. The side opposite this angle would be the chord length AB, but again, we don't need that since we're looking for the arc length.So, yes, sticking with the arc length formula is the right approach here.Therefore, I can confidently say that the spherical distance between points A and B is πR/6.**Final Answer**The spherical distance between points A and B is boxed{dfrac{pi R}{6}}.

answer：Alright, so I've got this probability problem here, and I need to figure out the chance that when I roll two dice, the resulting pair (a, b) falls into the intersection of two sets, A and B. Let me see if I can break this down step by step.First, let me understand what sets A and B are. Set A is defined as all points (x, y) where y is greater than or equal to the absolute value of (x - 1). So, mathematically, that's y ≥ |x - 1|. Set B is all points (x, y) where y is less than or equal to negative absolute value of x plus 5, which is y ≤ -|x| + 5.Okay, so I need to find the intersection of these two sets, which means I'm looking for all points (x, y) that satisfy both inequalities simultaneously. Since we're dealing with dice rolls, both a and b are integers between 1 and 6. So, x can be 1 through 6, and y can be 1 through 6 as well.Let me visualize this. For Set A, y ≥ |x - 1|. This is a V-shaped graph that opens upwards with its vertex at (1, 0). For Set B, y ≤ -|x| + 5. This is another V-shaped graph, but it opens downwards with its vertex at (0, 5). The intersection of these two sets would be the region where both inequalities are satisfied.Since we're dealing with integer values from 1 to 6 for both x and y, I can approach this by checking each possible value of x (which is 'a' from the dice roll) and determining the corresponding valid y values (which is 'b') that satisfy both inequalities.Let me start by listing out the possible values for 'a' and then find the corresponding 'b's.1. When a = 1: - For Set A: y ≥ |1 - 1| = 0. Since y has to be at least 0, but since y is at least 1 (because dice rolls are 1-6), this simplifies to y ≥ 1. - For Set B: y ≤ -|1| + 5 = -1 + 5 = 4. - So, combining both, y must be between 1 and 4, inclusive. Therefore, b can be 1, 2, 3, or 4. That's 4 possible values.2. When a = 2: - For Set A: y ≥ |2 - 1| = 1. - For Set B: y ≤ -|2| + 5 = -2 + 5 = 3. - So, y must be between 1 and 3, inclusive. Therefore, b can be 1, 2, or 3. That's 3 possible values.3. When a = 3: - For Set A: y ≥ |3 - 1| = 2. - For Set B: y ≤ -|3| + 5 = -3 + 5 = 2. - So, y must be exactly 2. Therefore, b can only be 2. That's 1 possible value.4. When a = 4: - For Set A: y ≥ |4 - 1| = 3. - For Set B: y ≤ -|4| + 5 = -4 + 5 = 1. - Wait, this is a problem. y has to be both greater than or equal to 3 and less than or equal to 1. There's no integer y that satisfies both conditions. So, there are 0 possible values for b here.5. When a = 5: - For Set A: y ≥ |5 - 1| = 4. - For Set B: y ≤ -|5| + 5 = -5 + 5 = 0. - Again, y has to be at least 4 and at most 0, which is impossible. So, no possible values for b.6. When a = 6: - For Set A: y ≥ |6 - 1| = 5. - For Set B: y ≤ -|6| + 5 = -6 + 5 = -1. - Similarly, y needs to be at least 5 and at most -1, which is impossible. So, no possible values for b.Alright, so compiling the results:- a = 1: 4 values of b- a = 2: 3 values of b- a = 3: 1 value of b- a = 4, 5, 6: 0 values of b eachAdding these up: 4 + 3 + 1 = 8 valid (a, b) pairs.Since we're rolling two dice, there are a total of 6 x 6 = 36 possible outcomes. Each outcome is equally likely, so the probability is the number of favorable outcomes divided by the total number of outcomes.So, the probability is 8/36. Simplifying that fraction by dividing numerator and denominator by 4 gives 2/9.Wait, let me double-check my calculations to make sure I didn't miss anything.For a = 1: y ≥ 0 and y ≤ 4. Since y starts at 1, that's 1-4, which is 4 numbers. Correct.For a = 2: y ≥ 1 and y ≤ 3. That's 1, 2, 3. Correct.For a = 3: y ≥ 2 and y ≤ 2. So, only 2. Correct.For a = 4: y ≥ 3 and y ≤ 1. Impossible. Correct.Same logic applies for a = 5 and 6. So, total of 8 favorable outcomes.Yes, 8/36 simplifies to 2/9. So, the probability is 2/9.I think that's solid. I don't see any mistakes in my reasoning. So, the answer should be 2/9.**Final Answer**boxed{B}

answer：Okay, so I have this problem about a convex quadrilateral ABCD that's rotated 90 degrees counterclockwise around a point O, and it becomes A'B'C'D'. Then, we're supposed to take midpoints of these segments A'B, B'C, C'D, and D'A, which are P, Q, R, S respectively. The goal is to prove that PR is perpendicular to QS and that PR equals QS.Hmm, let me try to visualize this. So, ABCD is a convex quadrilateral, meaning all its interior angles are less than 180 degrees, and none of its sides cross each other. When we rotate it 90 degrees around point O, which is somewhere in the plane, we get another convex quadrilateral A'B'C'D'. Now, midpoints: P is the midpoint of A'B, Q is the midpoint of B'C, R is the midpoint of C'D, and S is the midpoint of D'A. So, these midpoints are halfway along each of these segments.I need to show that PR is perpendicular to QS, and that their lengths are equal. Hmm, okay. Maybe I can use coordinate geometry for this. Let me assign coordinates to the points.Let me consider point O as the origin in the complex plane. So, I can represent points A, B, C, D as complex numbers a, b, c, d. Then, when we rotate them 90 degrees counterclockwise around O, each point z becomes iz, right? Because multiplying by i in the complex plane gives a 90-degree rotation.So, A' is ia, B' is ib, C' is ic, D' is id. Got it.Now, let's find the coordinates of the midpoints P, Q, R, S.P is the midpoint of A'B, so its coordinate is (A' + B)/2, which is (ia + b)/2.Similarly, Q is the midpoint of B'C, so it's (B' + C)/2 = (ib + c)/2.R is the midpoint of C'D, so it's (C' + D)/2 = (ic + d)/2.S is the midpoint of D'A, so it's (D' + A)/2 = (id + a)/2.Alright, so now I have coordinates for P, Q, R, S in terms of a, b, c, d.Now, I need to find vectors PR and QS.Vector PR is R - P, which is [(ic + d)/2] - [(ia + b)/2] = (ic + d - ia - b)/2.Similarly, vector QS is S - Q, which is [(id + a)/2] - [(ib + c)/2] = (id + a - ib - c)/2.Hmm, let's write these vectors more neatly.PR = (d - b + i(c - a))/2QS = (a - c + i(d - b))/2Wait a minute, that's interesting. Let me see:If I factor out the 1/2, I can write PR as (d - b)/2 + i(c - a)/2.Similarly, QS is (a - c)/2 + i(d - b)/2.Hmm, so PR is (d - b + i(c - a))/2, and QS is (a - c + i(d - b))/2.Is there a relationship between these two vectors?Let me see. If I take PR and multiply it by i, what do I get?Multiplying PR by i: i * [(d - b) + i(c - a)] / 2 = [i(d - b) + i^2(c - a)] / 2 = [i(d - b) - (c - a)] / 2.Which is equal to [-(c - a) + i(d - b)] / 2 = (a - c + i(d - b))/2, which is exactly QS.So, QS = i * PR.That's a key insight. So, vector QS is equal to i times vector PR.In complex numbers, multiplying by i corresponds to a 90-degree rotation. So, this means that vector QS is vector PR rotated by 90 degrees. Therefore, QS is perpendicular to PR.Moreover, since multiplying by i doesn't change the length, the lengths of PR and QS are equal.So, that shows both that PR is perpendicular to QS and that their lengths are equal.Wait, let me double-check that. If QS = i * PR, then indeed, they are perpendicular because multiplying by i rotates the vector by 90 degrees, and their lengths are the same because the magnitude of i is 1.So, that seems solid.Alternatively, I could think about this using vectors in the plane. If two vectors are perpendicular, their dot product is zero. Let me verify that.Let me denote vector PR as (d - b, c - a)/2 and vector QS as (a - c, d - b)/2.Wait, actually, in the complex plane, the real part is the x-component and the imaginary part is the y-component.So, vector PR has components ( (d - b)/2, (c - a)/2 ).Vector QS has components ( (a - c)/2, (d - b)/2 ).So, their dot product is:[(d - b)/2 * (a - c)/2] + [(c - a)/2 * (d - b)/2]Which is [(d - b)(a - c) + (c - a)(d - b)] / 4But notice that (c - a) = -(a - c), so the second term becomes -(a - c)(d - b).Therefore, the numerator is (d - b)(a - c) - (a - c)(d - b) = 0.Hence, the dot product is zero, which confirms that PR is perpendicular to QS.Also, since QS is i times PR, their magnitudes are equal.Therefore, both PR perpendicular to QS and PR = QS are proven.I think that wraps it up.**Final Answer**boxed{PR perp QS text{ and } PR = QS}

answer：Alright, so I've got this problem here about a sequence defined by a recurrence relation, and I need to find the sum of the first 60 terms. Let me try to unpack this step by step.First, the problem gives me the initial terms: ( a_1 = 1 ) and ( a_2 = 1 ). Then, for ( n geq 3 ), the sequence is defined by the recurrence relation ( a_n = a_{n-1} + 2a_{n-2} + 1 ). I need to find ( S_{60} ), which is the sum of the first 60 terms of this sequence.Hmm, okay. So, the sequence starts with two 1s, and each subsequent term is the previous term plus twice the term before that, plus 1. That seems a bit complicated, but maybe I can find a pattern or a closed-form expression for ( a_n ).Let me write down the first few terms to see if I can spot a pattern.- ( a_1 = 1 )- ( a_2 = 1 )- ( a_3 = a_2 + 2a_1 + 1 = 1 + 2*1 + 1 = 4 )- ( a_4 = a_3 + 2a_2 + 1 = 4 + 2*1 + 1 = 7 )- ( a_5 = a_4 + 2a_3 + 1 = 7 + 2*4 + 1 = 16 )- ( a_6 = a_5 + 2a_4 + 1 = 16 + 2*7 + 1 = 31 )- ( a_7 = a_6 + 2a_5 + 1 = 31 + 2*16 + 1 = 64 )- ( a_8 = a_7 + 2a_6 + 1 = 64 + 2*31 + 1 = 127 )Wait a minute, these numbers look familiar. 1, 1, 4, 7, 16, 31, 64, 127... Hmm, 4 is 2 squared, 7 is one less than 8, which is 2 cubed, 16 is 2 to the 4th, 31 is one less than 32, which is 2 to the 5th, 64 is 2 to the 6th, and 127 is one less than 128, which is 2 to the 7th. So, it seems like ( a_n ) is alternating between powers of 2 and one less than powers of 2.Let me see:- ( a_1 = 1 = 2^0 )- ( a_2 = 1 = 2^0 )- ( a_3 = 4 = 2^2 )- ( a_4 = 7 = 2^3 - 1 )- ( a_5 = 16 = 2^4 )- ( a_6 = 31 = 2^5 - 1 )- ( a_7 = 64 = 2^6 )- ( a_8 = 127 = 2^7 - 1 )So, it seems like for odd ( n ), ( a_n = 2^{n-1} ), and for even ( n ), ( a_n = 2^{n-1} - 1 ). Let me test this hypothesis with the given recurrence relation.Let's assume that for some ( k geq 2 ), ( a_{2k-1} = 2^{2k-2} ) and ( a_{2k} = 2^{2k-1} - 1 ). Then, let's compute ( a_{2k+1} ) and ( a_{2k+2} ) using the recurrence relation.Compute ( a_{2k+1} = a_{2k} + 2a_{2k-1} + 1 ).Substituting the assumed values:( a_{2k+1} = (2^{2k-1} - 1) + 2*(2^{2k-2}) + 1 )Simplify:( = 2^{2k-1} - 1 + 2^{2k-1} + 1 )( = 2^{2k-1} + 2^{2k-1} )( = 2*2^{2k-1} )( = 2^{2k} )Which matches our assumption that ( a_{2k+1} = 2^{2k} ).Now, compute ( a_{2k+2} = a_{2k+1} + 2a_{2k} + 1 ).Substituting the assumed values:( a_{2k+2} = 2^{2k} + 2*(2^{2k-1} - 1) + 1 )Simplify:( = 2^{2k} + 2^{2k} - 2 + 1 )( = 2^{2k} + 2^{2k} - 1 )( = 2*2^{2k} - 1 )( = 2^{2k+1} - 1 )Which also matches our assumption that ( a_{2k+2} = 2^{2k+1} - 1 ).So, by induction, our hypothesis holds. Therefore, the general formula for ( a_n ) is:- If ( n ) is odd, ( a_n = 2^{n-1} )- If ( n ) is even, ( a_n = 2^{n-1} - 1 )Okay, now that I have a general formula for ( a_n ), I can proceed to find ( S_{60} ), the sum of the first 60 terms.Since 60 is an even number, let's consider pairing the terms. Each pair consists of an odd-indexed term and the following even-indexed term.So, pair 1: ( a_1 + a_2 = 1 + 1 = 2 )Pair 2: ( a_3 + a_4 = 4 + 7 = 11 )Pair 3: ( a_5 + a_6 = 16 + 31 = 47 )Pair 4: ( a_7 + a_8 = 64 + 127 = 191 )Wait, let's see if there's a pattern here. The sums of the pairs are 2, 11, 47, 191... Hmm, each time it seems like the sum is roughly quadrupling and then subtracting something.Wait, 2 to 11: 2*4 + 3 = 1111*4 + 3 = 4747*4 + 3 = 191191*4 + 3 = 767Wait, no, 2*4=8, 8+3=11; 11*4=44, 44+3=47; 47*4=188, 188+3=191; 191*4=764, 764+3=767. So, each pair sum is 4 times the previous pair sum plus 3.But I'm not sure if this is helpful. Maybe instead, let's express each pair in terms of the general formula.For pair ( k ), where ( k ) ranges from 1 to 30 (since 60 terms make 30 pairs), the terms are ( a_{2k-1} ) and ( a_{2k} ).Using the general formula:( a_{2k-1} = 2^{2k-2} )( a_{2k} = 2^{2k-1} - 1 )So, the sum of the pair is:( a_{2k-1} + a_{2k} = 2^{2k-2} + (2^{2k-1} - 1) )Simplify:( = 2^{2k-2} + 2^{2k-1} - 1 )( = 2^{2k-2} + 2*2^{2k-2} - 1 )( = 3*2^{2k-2} - 1 )So, each pair sum is ( 3*4^{k-1} - 1 ) because ( 2^{2k-2} = 4^{k-1} ).Therefore, the total sum ( S_{60} ) is the sum of all 30 pairs:( S_{60} = sum_{k=1}^{30} (3*4^{k-1} - 1) )Let's split this into two separate sums:( S_{60} = 3sum_{k=1}^{30} 4^{k-1} - sum_{k=1}^{30} 1 )Compute each sum separately.First, compute ( sum_{k=1}^{30} 4^{k-1} ). This is a geometric series with first term ( 4^{0} = 1 ), common ratio 4, and 30 terms.The sum of a geometric series is given by ( S = frac{a(r^n - 1)}{r - 1} ), where ( a ) is the first term, ( r ) is the common ratio, and ( n ) is the number of terms.So,( sum_{k=1}^{30} 4^{k-1} = frac{1*(4^{30} - 1)}{4 - 1} = frac{4^{30} - 1}{3} )Next, compute ( sum_{k=1}^{30} 1 ). This is simply 30, since we're adding 1 thirty times.Putting it all together:( S_{60} = 3*left(frac{4^{30} - 1}{3}right) - 30 )Simplify:( S_{60} = (4^{30} - 1) - 30 )( S_{60} = 4^{30} - 31 )So, the sum of the first 60 terms is ( 4^{30} - 31 ), which corresponds to option B.

answer：Okay, so I have this problem where I need to find the inverse function of g(x) at 1/4. The function g(x) is given as (x³ - 3)/2. Hmm, inverse functions can be a bit tricky, but I think I remember that to find g⁻¹(1/4), I need to find the value of x such that g(x) equals 1/4. Alright, let me write that down. So, if g(x) = 1/4, then (x³ - 3)/2 = 1/4. I need to solve this equation for x. First, maybe I should get rid of the denominator to make it easier. If I multiply both sides of the equation by 2, that should clear the fraction. Let's see: 2*(x³ - 3)/2 = 2*(1/4). Simplifying that, the 2s on the left side cancel out, leaving me with x³ - 3. On the right side, 2*(1/4) is 1/2. So now I have x³ - 3 = 1/2.Next, I want to isolate x³, so I'll add 3 to both sides of the equation. That gives me x³ = 1/2 + 3. Wait, 3 is the same as 6/2, right? So 1/2 + 6/2 is 7/2. Therefore, x³ = 7/2.Now, to solve for x, I need to take the cube root of both sides. The cube root of x³ is just x, and the cube root of 7/2 is... hmm, I don't think that simplifies nicely. So, x equals the cube root of 7/2. Let me double-check my steps to make sure I didn't make a mistake. Starting with g(x) = 1/4, I set up the equation (x³ - 3)/2 = 1/4. Multiplying both sides by 2 gives x³ - 3 = 1/2. Adding 3 to both sides, which is 6/2, gives x³ = 7/2. Taking the cube root of both sides, x = ∛(7/2). I think that makes sense. I don't see any errors in my calculations. So, the inverse function g⁻¹(1/4) is equal to the cube root of 7/2. I guess that's the answer. It doesn't look like it can be simplified further, so I'll leave it like that.**Final Answer**The value of ( g^{-1}left(frac{1}{4}right) ) is boxed{sqrt[3]{frac{7}{2}}}.

answer：Alright, so I have this problem where a store buys goods from a manufacturer at 20 yuan per item. They want to set a retail price, p yuan, and the sales volume, Q, depends on this price. The relationship is given by Q = 8300 - 170p - p². I need to figure out what price p will maximize the gross profit L, and also find out what that maximum gross profit is.Okay, first, I need to understand what gross profit is. I think gross profit is the total revenue minus the cost. So, if they sell Q items at p yuan each, their revenue is p times Q. Their cost is the amount they paid to the manufacturer, which is 20 yuan per item, so that's 20 times Q. Therefore, the gross profit L should be:L = Revenue - Cost = pQ - 20QWhich simplifies to:L = Q(p - 20)Now, I can substitute the given Q into this equation. So, Q is 8300 - 170p - p². Therefore:L = (8300 - 170p - p²)(p - 20)Hmm, I need to expand this to make it easier to work with. Let me multiply it out.First, multiply 8300 by (p - 20):8300 * p = 8300p8300 * (-20) = -166000Next, multiply -170p by (p - 20):-170p * p = -170p²-170p * (-20) = +3400pThen, multiply -p² by (p - 20):-p² * p = -p³-p² * (-20) = +20p²Now, let's add all these terms together:8300p - 166000 - 170p² + 3400p - p³ + 20p²Combine like terms:- p³ (only one term)-170p² + 20p² = -150p²8300p + 3400p = 11700p-166000 (constant term)So, putting it all together:L = -p³ - 150p² + 11700p - 166000Alright, now I have the gross profit function in terms of p. To find the maximum gross profit, I need to find the value of p that maximizes L. Since this is a cubic function, and the coefficient of p³ is negative, the function will have a maximum point.To find the maximum, I should take the derivative of L with respect to p and set it equal to zero. This will give me the critical points where the function could have a maximum or minimum.So, let's compute the derivative L'(p):L'(p) = dL/dp = -3p² - 300p + 11700Now, set L'(p) = 0 to find the critical points:-3p² - 300p + 11700 = 0Hmm, this is a quadratic equation. Let's simplify it by dividing all terms by -3 to make it easier:p² + 100p - 3900 = 0Now, I can solve this quadratic equation using the quadratic formula:p = [-b ± √(b² - 4ac)] / 2aWhere a = 1, b = 100, and c = -3900.Plugging in the values:p = [-100 ± √(100² - 4*1*(-3900))] / 2*1Calculate the discriminant:100² = 100004*1*(-3900) = -15600So, the discriminant is:10000 - (-15600) = 10000 + 15600 = 25600Now, take the square root of 25600:√25600 = 160So, the solutions are:p = [-100 ± 160] / 2First solution:p = (-100 + 160) / 2 = 60 / 2 = 30Second solution:p = (-100 - 160) / 2 = (-260) / 2 = -130Well, a price of -130 yuan doesn't make sense in this context, so we can discard that. Therefore, the critical point is at p = 30 yuan.Now, to ensure that this is indeed a maximum, I should check the second derivative or analyze the behavior of the first derivative around p = 30.Let's compute the second derivative L''(p):L''(p) = d/dp [L'(p)] = d/dp [-3p² - 300p + 11700] = -6p - 300Now, evaluate L''(p) at p = 30:L''(30) = -6*30 - 300 = -180 - 300 = -480Since L''(30) is negative, this indicates that the function is concave down at p = 30, which means that p = 30 is indeed a local maximum.Therefore, the price that maximizes gross profit is 30 yuan.Now, to find the maximum gross profit, I need to substitute p = 30 back into the gross profit function L(p):L(30) = -30³ - 150*(30)² + 11700*30 - 166000Let's compute each term step by step:First, -30³:30³ = 27000So, -27000Next, -150*(30)²:30² = 900150*900 = 135000So, -135000Then, 11700*30:11700*30 = 351000Lastly, -166000Now, add all these together:-27000 - 135000 + 351000 - 166000Let's compute step by step:-27000 - 135000 = -162000-162000 + 351000 = 189000189000 - 166000 = 23000So, L(30) = 23000 yuanTherefore, the maximum gross profit is 23000 yuan when the price is set at 30 yuan per item.Just to double-check, let's ensure that our calculations are correct.First, when we expanded L(p), we had:L = (8300 - 170p - p²)(p - 20)Expanding this:8300p - 166000 - 170p² + 3400p - p³ + 20p²Combining like terms:-p³ - 150p² + 11700p - 166000That seems correct.Taking the derivative:L'(p) = -3p² - 300p + 11700Setting to zero:-3p² - 300p + 11700 = 0Dividing by -3:p² + 100p - 3900 = 0Quadratic formula:p = [-100 ± √(10000 + 15600)] / 2 = [-100 ± 160] / 2Which gives p = 30 or p = -130Discarding the negative price, p = 30.Second derivative:L''(p) = -6p - 300At p = 30, L''(30) = -180 - 300 = -480 < 0, so it's a maximum.Calculating L(30):-30³ = -27000-150*(30)² = -150*900 = -13500011700*30 = 351000-166000Adding them up:-27000 - 135000 = -162000-162000 + 351000 = 189000189000 - 166000 = 23000Yes, that seems consistent.So, all steps check out, and the calculations are correct.