Question

Revenue is given by R(q) = 450q and cost is given by C(q) = 10,000 + 3q². At what quantity is profit maximized? What is the total profit at his production level?

Answer 1

To maximise profit, calculate the quantity where total revenue exceeds total cost by the largest amount using derivatives of profit function and apply the second derivative test. Then, calculate the total profit by evaluating the profit function at the optimal output level.

Step-by-step explanation:

The student is asking how to calculate the quantity of output that will provide the highest level of profit. To find the quantity where profit is maximized, we need to determine the point where total revenue (TR) exceeds total cost (TC) by the largest margin.

Given the functions R(q) = 450q and C(q) = 10,000 + 3q², we first need to calculate the profit function P(q) by subtracting the total cost from the total revenue, which is P(q) = 450q - (10,000 + 3q²). The next step is to take the derivative of the profit function and set it equal to zero to find the quantity q that maximizes profit. Then, we check if this q gives us the maximum profit by ensuring the second derivative is negative. Finally, we calculate the total profit at this production level by inserting the optimal quantity into the profit function.

Without the actual derivative calculations and second derivative test, we cannot specify the exact quantity. However, the student can apply these steps to maximize profit and calculate the total profit at that production level.