using the cost function C(q) = q3−60q2+1200q+760 for 0 ≤ q ≤ 50 and a price per unit of $551.
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 = $
revenue is price * quantity, so
R(q) = 551q
profit is revenue minus cost, so
P(q) = 551q - (q^3-60q^2+1200q+760)
You should probably graph P(q) to find the max. Take a look here:
http://www.wolframalpha.com/input/?i=551q+-+%28q^3-60q^2%2B1200q%2B760%29
If you have calculus as a tool, just find where dP/dq=0
To find the production level that maximizes profit, we need to find the point where the difference between revenue and cost is maximized.
a) To find the production level that maximizes profit, we need to find the value of q that corresponds to the maximum value of the profit function. The profit function is given by:
Profit(q) = Revenue(q) - Cost(q)
Given the cost function C(q) = q^3 - 60q^2 + 1200q + 760 and the price per unit of $551, the revenue function is:
Revenue(q) = Price per unit * q = 551q
To find the production level that maximizes profit, we need to maximize the function Profit(q) = Revenue(q) - Cost(q). This can be done by finding the critical points of the profit function and determining which critical point gives the maximum profit.
First, calculate the derivative of the profit function:
Profit'(q) = Revenue'(q) - Cost'(q)
Taking the derivatives, we get:
Profit'(q) = 551 - (3q^2 - 120q + 1200)
Now set the derivative equal to zero and solve for q to find the critical points:
551 - (3q^2 - 120q + 1200) = 0
Rearranging, we have:
3q^2 - 120q + 649 = 0
To solve this quadratic equation, you can use the quadratic formula:
q = (-b ± sqrt(b^2 - 4ac)) / 2a
Substituting the coefficients from the quadratic equation, we have:
q = (120 ± sqrt(120^2 - 4(3)(649))) / (2(3))
Calculate the solutions for q using this formula. The two solutions for q will give two critical points. To determine which critical point gives the maximum profit, evaluate the profit function at both critical points. The value of q that gives the maximum profit will be the answer to part a).
b), c), d) Once you have the value of q that maximizes profit (obtained in part a)), substitute this value into the cost function C(q) = q^3 - 60q^2 + 1200q + 760 to find the total cost at this production level.
Similarly, substitute this value of q into the revenue function Revenue(q) = 551q to find the total revenue at this production level.
Finally, subtract the total cost from the total revenue to find the total profit at this production level. Round all the answers to the nearest whole number.