The demand function for a product is p=20−5q where p is the price in dollars when q units are demanded.
Find the level of production that maximizes the total revenue and determine the revenue. q= units R= $
To find the level of production that maximizes total revenue, we need to find the quantity at which the derivative of the revenue function with respect to q is equal to zero.
Total Revenue (R) can be calculated by multiplying the price (p) by the quantity (q):
R = p * q
Given the demand function p = 20 - 5q, we can substitute this into the revenue function:
R = (20 - 5q) * q
Now, let's calculate the derivative of the revenue function with respect to q:
dR/dq = 20q - 5q^2
To find the maximum point, we set the derivative equal to zero and solve for q:
20q - 5q^2 = 0
Factoring out q, we get:
q(20 - 5q) = 0
Setting each factor equal to zero and solving for q gives us two potential values for q:
1) q = 0
2) 20 - 5q = 0
5q = 20
q = 4
Since we're interested in the quantity of production, q, it cannot be zero (as it represents no production). Therefore, q = 4 is the level of production that maximizes total revenue.
Next, we need to find the revenue at this level of production. We can substitute q = 4 into the revenue function:
R = (20 - 5q) * q
R = (20 - 5*4) * 4
R = (20 - 20) * 4
R = 0 * 4
R = 0
Therefore, at a production level of 4 units, the total revenue is $0.