The demand function for a good is p=40−0.02q. 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 = $
assuming q is the quantity, p is the price,
(a) Revenue r(q) = price * demand = q*p(q)
I guess "production level" is supposed to be the quantity demanded ...
That would make
(b) find q where dr/dq = 0
(c,d) evaluate p(q),r(q) there
a) To find the revenue as a function of production, we multiply the price (p) by the quantity (q). The revenue function (R(q)) is given by the equation:
R(q) = p * q
Substituting the given demand function into this equation, we have:
R(q) = (40 - 0.02q) * q
Simplifying further:
R(q) = 40q - 0.02q^2
Therefore, the revenue function as a function of production, q, is:
R(q) = 40q - 0.02q^2
b) To maximize revenue, we need to find the production level (q) at which the derivative of the revenue function with respect to q is equal to zero. Taking the derivative of R(q), we have:
dR(q)/dq = 40 - 0.04q
Setting this derivative equal to zero and solving for q:
40 - 0.04q = 0
0.04q = 40
q = 40 / 0.04
q = 1000
Therefore, the production level that maximizes revenue is q = 1000.
c) To find the price corresponding to this production level, we substitute q = 1000 into the demand function:
p = 40 - 0.02q
p = 40 - 0.02 * 1000
p = 40 - 20
p = 20
Therefore, the price corresponding to the production level of q = 1000 is price = $20.
d) Finally, to find the total revenue at this production level, we substitute q = 1000 into the revenue function:
R(q) = 40q - 0.02q^2
R(1000) = 40 * 1000 - 0.02 * (1000^2)
R(1000) = 40000 - 0.02 * 1000000
R(1000) = 40000 - 20000
R(1000) = 20000
Therefore, the total revenue at the production level q = 1000 is revenue = $20000.
a) To write the revenue as a function of production, q, we need to multiply the quantity sold by the price per unit. The quantity sold is q, and the price per unit is p. Therefore, Revenue (R) is given by:
R(q) = q * p
We are given the demand function p = 40 - 0.02q. Substituting this into the revenue equation, we get:
R(q) = q * (40 - 0.02q)
b) To find the production level that maximizes revenue, we need to find the value of q that maximizes R(q). One way to do this is by finding the vertex of the quadratic equation R(q) = q * (40 - 0.02q). Alternatively, we can find the derivative of R(q) with respect to q and set it equal to zero.
c) To determine the price corresponding to the production level that maximizes revenue, we substitute the value of q found in part (b) into the demand function p = 40 - 0.02q. This will give us the value of the price at that production level.
d) To find the total revenue at the production level that maximizes revenue, we substitute the value of q found in part (b) into the revenue function R(q) = q * (40 - 0.02q). This will give us the total revenue generated at that production level.