Final answer:
To find the marginal revenue when profits are $15000, solve for the quantity (q) that corresponds to a profit of $15000 using the profit function. The quantity is approximately 349.99. Then, calculate the derivative of the revenue function with respect to q and substitute the value of q to find the marginal revenue. The marginal revenue is approximately $9.06.
Step-by-step explanation:
To find the marginal revenue when profits are $15000, we need to determine the quantity (q) that corresponds to a profit of $15000. We can use the profit function, which is given by:
Profit = Revenue - Cost
Since Profit is given as $15000, we can set up the equation:
15000 = p(q) - C(q)
Substituting the given cost and price functions, we have:
15000 = (-2.7q + 950) - (110q + 45000)
Simplifying the equation, we get:
15000 = -112.7q + 54450
112.7q = 54450 - 15000
112.7q = 39450
q = 39450 / 112.7
q ≈ 349.99
To find marginal revenue, we can calculate the derivative of the revenue function. The revenue function is:
R(q) = p(q) * q
Taking the derivative of R(q) with respect to q, we get:
R'(q) = p'(q) * q + p(q)
Substituting the given price function, we have:
R'(q) = -2.7 * q + 950 + (-2.7q + 950)
Simplifying the equation, we get:
R'(q) = -5.4q + 1900
Substituting the value of q we found earlier, we have:
R'(349.99) ≈ -5.4(349.99) + 1900
R'(349.99) ≈ -1890.94 + 1900
R'(349.99) ≈ 9.06
Therefore, the marginal revenue when profits are $15000 is approximately $9.06.