Question

Widgets, Inc., has determined that its demand function is p=40−4q. It also knows that its cost function is C(q)=2q. What is the firm's maximum profit?

Note: Enter your answer as a number only, without any special characters. For example, if you determined that the answer is $1,000, enter this as follows: 1000
Bob invests 51000 in a fund that refurns 6% compounded continuously. How long will it take for Bobtoo accumulate s3000? Note: Enter your answer as a number only, without any special characters or words. Round your answer to the whole year. For example, if you determined that the answer is 6.11 years enter this as follows: 6

Answer 1

To calculate Widgets, Inc.'s maximum profit, the profit function derived from their demand and cost functions is maximized, resulting in a maximum profit of 90.25 when output (q) is 4.75.

Step-by-step explanation:

To determine the maximum profit for Widgets, Inc., we need to find the level of output (q) that maximizes profits given the demand function (p=40-4q) and the cost function (C(q)=2q).

Profit function (π) is defined as total revenue (TR) minus total costs (TC). Total revenue can be calculated by multiplying the price (p) by the quantity (q), so TR = pq. First, we insert the demand function into the TR equation to get TR = q(40-4q) = 40q - 4q². Total cost is given by the cost function, TC = C(q) = 2q.

The profit function is thus π(q) = TR - TC = (40q - 4q²) - 2q. Simplifying the profit function gives us π(q) = 40q - 4q² - 2q = 38q - 4q². To find the value of q that maximizes profit, we take the derivative of the profit function with respect to q and set it equal to zero: dπ/dq = 38 - 8q = 0, which when solved gives us q=4.75.

Inserting q=4.75 back into the profit function, we get maximum profit π(4.75) = 38(4.75) - 4(4.75)² = 180.50 - 90.25 = 90.25.