using the demand function p = 41 − 0.04 q instead of the one given in your book. Round your numeric answers to one decimal place.
a) Write the revenue as a function of production, q.
R(q) =
b) What production level maximizes revenue?
q =
c) What price corresponds to this production level?
price = $
d) What is the total revenue at this production level?
revenue = $
a) To write the revenue as a function of production, q, we can use the formula: R(q) = p * q, where p represents the price determined by the demand function.
Given the demand function p = 41 - 0.04q, we substitute the value of p into the revenue formula:
R(q) = (41 - 0.04q) * q
Rearranging and simplifying the equation:
R(q) = 41q - 0.04q^2
Therefore, the revenue function is R(q) = 41q - 0.04q^2.
b) To find the production level that maximizes revenue, we need to determine the value of q that gives the maximum value for R(q).
To find the maximum for R(q), we take the derivative of R(q) with respect to q and set it equal to zero. This will help us find the critical points (where the slope is 0).
R'(q) = 41 - 0.08q
Setting R'(q) = 0:
0 = 41 - 0.08q
Rearranging the equation:
0.08q = 41
q = 41 / 0.08
q = 512.5
Therefore, the production level that maximizes revenue is q = 512.5.
c) To find the corresponding price at this production level, substitute the value of q into the demand function p = 41 - 0.04q:
p = 41 - 0.04 * 512.5
p = 41 - 20.5
p = 20.5
Therefore, the price corresponding to the production level q = 512.5 is $20.5.
d) To calculate the total revenue at this production level, substitute the value of q into the revenue function R(q) = 41q - 0.04q^2:
revenue = R(512.5) = 41 * 512.5 - 0.04 * (512.5)^2
revenue = 21,093.75 - 105,156.25
revenue = -84,062.5
Therefore, the total revenue at the production level q = 512.5 is -$84,062.5.