The cost function for a product is C(q)=q^3−63q^2+1323q+920 for 0≤q≤50 and a price per unit of $515.
Round your answers to the nearest whole number.
a) What production level maximizes profit?
q =
b) What is the total cost at this production level?
cost = $
c) What is the total revenue at this production level?
revenue = $
d) What is the total profit at this production level?
profit = $
as I said, revenue = price * quantity = 515q = 515*34 = 17,510
revenue is price * quantity, so
r(q) = 515q
profit = revenue - cost, so
p(q) = 515q - (q^3−63q^2+1323q+920)
(a) where does p'(q) = 0?
(b) c(q) where q is the solution to (a)
(c) and (d) should be clear
I got all except C. That is the only one I am struggling with.
A) q=34
B) $12378
C)
D) $5132
To find the production level that maximizes profit, we need to determine the derivative of the profit function with respect to q and set it equal to zero. Let's go through the steps:
Step 1: Find the profit function.
The profit function is given by P(q) = R(q) - C(q), where R(q) is the revenue function and C(q) is the cost function. In this case, the profit function is:
P(q) = (Selling Price per Unit) * (q) - C(q)
P(q) = 515 * q - (q^3 - 63q^2 + 1323q + 920)
Step 2: Take the derivative of the profit function.
To find the derivative, we differentiate the profit function with respect to q. In this case, we can simplify the profit function as follows:
P(q) = 515q - q^3 + 63q^2 - 1323q - 920
P'(q) = 515 - 3q^2 + 126q - 1323
Step 3: Set the derivative equal to zero and solve for q.
To find the production level that maximizes profit, we set the derivative equal to zero and solve for q:
515 - 3q^2 + 126q - 1323 = 0
Step 4: Solve the equation for q.
To solve this equation, we can rearrange it as follows:
-3q^2 + 126q - 808 = 0
We can use factoring or the quadratic formula to solve this quadratic equation. After finding the values of q, we select the one that lies within the range 0 ≤ q ≤ 50 since it represents the production level.
Once we have the value of q, we can substitute it back into the cost function to find the total cost (b), into the revenue function to find the total revenue (c), and subtract the total cost from the total revenue to find the total profit (d).
Please solve the quadratic equation to find the production level that maximizes profit (a), and then let me know the value you obtain so I can help you further with parts (b), (c), and (d).