Rite-Cut riding lawnmowers obey the demand equation p = -1/20x + 940. The cost o

Question

Rite-Cut riding lawnmowers obey the demand equation p = -1/20x + 940. The cost of producing x lawnmowers is given by the function C(x)=140x + 6000. Express the revenue R as a function of x. Express the profit P as a function of x. Find the value of x that maximizes profit. What is the maximum profit? What price should be charged in order to maximize profit? What quantity will maximize the profit? lawnmowers What is the maximum profit? $ (Round to the nearest dollar as needed.) What price should be charged for the maximum profit? $ (Round to the nearest cent as needed.) Enter your answer in each of the answer boxes

Explanation / Answer

a. Since the Rite- Cut riding lawnmowers obey the demand equation p = -x/20+940 and since C(x) = 140x+6000, hence the revenue function R(x) = price*quantity sold = p*x = (-x/20+940)x = -x2/20 +940x.

b. The profit function P(x) is R(x)-C(x) = -x2/20 +940x- 140x -6000 = -x2/20 + 800x -6000.

c. When P(x) is maximum, dP/dx = 0 and d2P/dx2 will be negative. Here, dP/dx = -2x/20 +800 = 800-x/10. Thus dP/dx = 0 when x/10 = 800 or, x = 10*800 = 8000. Also, d2P/dx2 = -2/20 = -1/10. Thus, P(x) is maximum when x = 8000 lawnmowers.

The maximum profit is the value of P(x) when x = 8000 i.e. –(8000)2/20 +800*8000 -6000 = -3200000+ 6400000- 6000 = $ 3194000.

d. Since p = -x/20 +940, the price that should be charged for maximum profit is -8000/20 +940 = -400+940 = $ 540.

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