The revenue and cost functions, both in dollars, for the production and sale of x TVs are given as R(x) = 160x − 0.8x2 and C(x) = 3,240 + 15x, respectively.(a)Find the value(s) of x where the graph of R(x) has a horizontal tangent line with correct units. (Enter your answers as a comma-separated list. Round your answers to the nearest integer. If an answer does not exist, enter DNE.)x = 100 (b)Find the profit function in terms of x. (Note: Profit is revenue minus costs.)P(x) = −.8x2+145x−3240 (c)Find the value(s) of x where the graph of P(x) has a horizontal tangent line with correct units. (Enter your answers as a comma-separated list. Round your answers to the nearest integer. If an answer does not exist, enter DNE.)x = 7258
Question
The revenue and cost functions, both in dollars, for the production and sale of x TVs are given as R(x) = 160x − 0.8x2 and C(x) = 3,240 + 15x, respectively.(a)Find the value(s) of x where the graph of R(x) has a horizontal tangent line with correct units. (Enter your answers as a comma-separated list. Round your answers to the nearest integer. If an answer does not exist, enter DNE.)x = 100 (b)Find the profit function in terms of x. (Note: Profit is revenue minus costs.)P(x) = −.8x2+145x−3240 (c)Find the value(s) of x where the graph of P(x) has a horizontal tangent line with correct units. (Enter your answers as a comma-separated list. Round your answers to the nearest integer. If an answer does not exist, enter DNE.)x = 7258
Solution
(a) The graph of R(x) has a horizontal tangent line where the derivative of R(x) is equal to zero. The derivative of R(x) = 160x - 0.8x^2 is R'(x) = 160 - 1.6x. Setting this equal to zero gives 160 - 1.6x = 0, which simplifies to x = 100. So, the graph of R(x) has a horizontal tangent line at x = 100.
(b) The profit function P(x) is given by the revenue function R(x) minus the cost function C(x). So, P(x) = R(x) - C(x) = (160x - 0.8x^2) - (3240 + 15x) = -0.8x^2 + 145x - 3240.
(c) The graph of P(x) has a horizontal tangent line where the derivative of P(x) is equal to zero. The derivative of P(x) = -0.8x^2 + 145x - 3240 is P'(x) = -1.6x + 145. Setting this equal to zero gives -1.6x + 145 = 0, which simplifies to x = 90.625. Rounding to the nearest integer gives x = 91. So, the graph of P(x) has a horizontal tangent line at x = 91.
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