Questions: The cost function for a product is C(q)=q^3-54q^2+972q+1180 for 0 ≤ q ≤ 50 and a price per unit of 526. 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 =

Solution Steps

To solve this problem, we need to determine the production level that maximizes profit, which is the difference between total revenue and total cost.

a) To find the production level that maximizes profit, we first need to calculate the profit function, which is the revenue function minus the cost function. The revenue function is given by the price per unit times the quantity. We then find the derivative of the profit function and solve for the quantity where the derivative is zero to find the critical points. We evaluate these points to find the maximum profit within the given range.

b) Once we have the production level that maximizes profit, we substitute this quantity into the cost function to find the total cost.

c) Similarly, we substitute the production level into the revenue function to find the total revenue.

The profit function \( P(q) \) is defined as the difference between total revenue \( R(q) \) and total cost \( C(q) \): \[ P(q) = R(q) - C(q) = 526q - (q^3 - 54q^2 + 972q + 1180) \] This simplifies to: \[ P(q) = -q^3 + 54q^2 - 446q - 1180 \]

To find the production level that maximizes profit, we take the derivative of the profit function and set it to zero: \[ P'(q) = -3q^2 + 108q - 446 \] Setting \( P'(q) = 0 \) gives us the critical points: \[ q = 18 - \frac{\sqrt{1578}}{3} \quad \text{and} \quad q = 18 + \frac{\sqrt{1578}}{3} \]

We evaluate the profit function at the critical points and the endpoints \( q = 0 \) and \( q = 50 \):

• At \( q = 0 \): \[ P(0) = -1180 \]
• At \( q = 50 \): \[ P(50) = -13480 \]
• At \( q = 18 - \frac{\sqrt{1578}}{3} \) and \( q = 18 + \frac{\sqrt{1578}}{3} \), we find the corresponding profit values.

The maximum profit occurs at: \[ q = 18 + \frac{\sqrt{1578}}{3} \]

Now we calculate the total cost \( C \) and total revenue \( R \) at the production level \( q = 18 + \frac{\sqrt{1578}}{3} \):

• Total Cost: \[ C\left(18 + \frac{\sqrt{1578}}{3}\right) = \left(18 + \frac{\sqrt{1578}}{3}\right)^3 - 54\left(18 + \frac{\sqrt{1578}}{3}\right)^2 + 972\left(18 + \frac{\sqrt{1578}}{3}\right) + 1180 \]
• Total Revenue: \[ R\left(18 + \frac{\sqrt{1578}}{3}\right) = 526\left(18 + \frac{\sqrt{1578}}{3}\right) \]

Final Answer