A company decides to begin making and selling computers. The price function is given as follows:
p=−70x+4000,
where x is the number of computers that can be sold at a price of p dollars per unit. Additionally, the financial department has determined that the weekly fixed cost of production will be 3000 dollars with an additional cost of 250 dollars per unit.
(A) Find the revenue function in terms of x.
R(x) =
(B) Use the financial department's estimates to determine the cost function in terms of x.
C(x) =
(C) Find the profit function in terms of x.
P(x) =
(D) Evaluate the marginal profit at x=250.
P′(250) =
The first is incorrect.
R(x) = x(-70x+4000) = -70x^2 +4000x
P(x) = (-70x^2 +4000x) - (3000+250x) = -70x^2 + 3750x - 3000
Dunno what the first person was thinking but these should be correct. :)
Oh and for the last part just get the derivative for P(x)
P'(x)= -140x + 3750
Then plug in 250 for x
P'(250)= -140(250) +3750 = -31250
A) The revenue function can be found by multiplying the price (p) by the number of units sold (x):
R(x) = px
Using the given price function p = -70x + 4000, we substitute this into the revenue function:
R(x) = (-70x + 4000)x
Simplifying further:
R(x) = -70x^2 + 4000x
B) The cost function includes both fixed costs and variable costs.
The fixed cost (FC) is given as $3000 per week.
The variable cost (VC) is given as $250 per unit.
So the cost function (C(x)) can be expressed as:
C(x) = FC + VC * x
C(x) = 3000 + 250x
C) The profit function (P(x)) is the difference between the revenue function (R(x)) and the cost function (C(x)):
P(x) = R(x) - C(x)
Substituting the previously determined revenue and cost functions:
P(x) = (-70x^2 + 4000x) - (3000 + 250x)
Simplifying:
P(x) = -70x^2 + 4000x - 3000 - 250x
P(x) = -70x^2 + 3750x - 3000
D) To evaluate the marginal profit at x = 250, we need to find the derivative of the profit function P(x) with respect to x, and then substitute x = 250 into the derivative expression.
P'(x) = -140x + 3750
Substituting x = 250:
P'(250) = -140(250) + 3750
P'(250) = -35000 + 3750
P'(250) = -31250
Therefore, the marginal profit at x = 250 is -31250 dollars.
r(x) = -70x+4000
c(x) = 3000 + 250x
p(x) = r(x)-c(x) = -320x