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 find the revenue as a function of production, q, you can multiply the quantity, q, by the price, p. The given demand function is p = 41 - 0.04q, so the revenue function can be calculated by substituting this value of p into the formula:
R(q) = q * p = q * (41 - 0.04q)
Therefore, the revenue function as a function of production, q, is R(q) = 41q - 0.04q^2.
b) To find the production level that maximizes revenue, we can take the derivative of the revenue function with respect to q and set it equal to zero to find the critical point:
R'(q) = 41 - 0.08q = 0
Solving this equation for q, we get:
41 - 0.08q = 0
0.08q = 41
q = 41 / 0.08
So the production level that maximizes revenue is q = 512.5.
c) To find the price corresponding to this production level, we can substitute q = 512.5 into the given demand function:
p = 41 - 0.04(512.5)
Simplifying this, we get:
p = 41 - 20.5
Therefore, the price corresponding to the production level that maximizes revenue is p = $20.5.
d) To find the total revenue at this production level, we can substitute q = 512.5 into the revenue function:
R(512.5) = 41(512.5) - 0.04(512.5)^2
Calculating this, we get:
R(512.5) = 21062.5 - 10506.25
Therefore, the total revenue at the production level that maximizes revenue is $10,556.25.