Question

Rite-Cut riding lawnmowers obey the demand equation p= -20X+910. The cost of

producing x lawnmowers is given by the function C(x) = 90x+6000.
a) Express the revenue R as a function of x.
b) Express the profit P as a function of x.
c) Find the value of x that maximizes profit. What is the maximum profit?
d) What price should be charged in order to maximize profit?
(Simplify your answer. Do not factor.)
(Simplify your answer. Do not factor.)
a) R(x)=
b) P(x) =
c) What quantity will maximize the profit?
lawnmowers
13

Answer 1

a) Revenue = Price * Quantity. So, R(x) = x * p = x*(-20x + 910) = -20x^2 + 910x.

b) Profit = Revenue - Cost. So, P(x) = R(x) - C(x) = (-20x^2 + 910x) - (90x + 6000) = -20x^2 + 820x - 6000.

c) To find the value of x that maximizes profit, we need to find the vertex of the quadratic function -20x^2 + 820x - 6000. The x-coordinate of the vertex is given by -b/(2a), where a = -20 and b = 820. So, x = -820/(2*(-20)) = 20.5. Since this is a parabola with a negative coefficient of x^2, the vertex represents a maximum. Therefore, the value of x that maximizes profit is 20.5, and the maximum profit is P(20.5) = -20(20.5)^2 + 820(20.5) - 6000 = $8,405.

d) To find the price that should be charged in order to maximize profit, we need to find the corresponding value of p when x = 20.5. Using the demand equation p = -20x + 910, we get p = -20(20.5) + 910 = $310. Therefore, the price that should be charged in order to maximize profit is $310.