The total cost
C(q)
of producing q goods is given by the following equation.
C(q) = 0.01q^3 − 0.6q^2 + 13q
a) What is the fixed Cost?
b) What is the maximum profit if each item is sold for $9? (Assume you sell everything you produce. Round your answer to the nearest cent.)
c) Suppose exactly 36 goods are produced. They all sell when the price is $9 each, but for each $1 increase in price, 2 fewer goods are sold. Should the price be raised?
^ I know that the price should be raised.
If the price should be raised, by how much?
^I don't know how to solve this problem.
To answer these questions, we need to understand the components of the cost function and use calculus concepts. Let's break it down step by step.
a) Finding the fixed cost:
To find the fixed cost, we need to identify the term that does not rely on the quantity of goods produced (q). In the given equation, only the constant term (13q) depends on q, while the other two terms involve powers of q. Thus, there is no fixed cost in this equation, as the cost changes with the quantity of goods produced.
b) Finding the maximum profit:
To find the maximum profit, we need to differentiate the cost function with respect to q to get the marginal cost function (which represents the rate of change of cost with respect to quantity). Then, we equate the marginal cost to the selling price (since profit is the difference between revenue and cost).
The marginal cost function is the derivative of the cost function:
C'(q) = 0.03q^2 - 1.2q + 13
Now, equate the marginal cost to the selling price:
0.03q^2 - 1.2q + 13 = 9
To find the values of q that satisfy this equation, we need to solve it by rearranging it to the quadratic form:
0.03q^2 - 1.2q + 4 = 0
To solve this quadratic equation for q, you can either use the quadratic formula or factorization. Once you solve the equation, you will get two values for q. The maximum profit corresponds to the quantity of goods that maximizes the profit, so choose the value of q that corresponds to the maximum profit.
To calculate the maximum profit, substitute the value of q obtained into the cost function:
C(q) = 0.01q^3 - 0.6q^2 + 13q
c) Should the price be raised? If so, by how much?
To answer this question, we need to analyze the relationship between the quantity of goods produced, the selling price, and the demand. From the information given, we know that when the selling price is $9, all 36 goods are sold. However, for each $1 increase in price, 2 fewer goods are sold.
To determine whether the price should be raised, consider the marginal revenue, which is the derivative of the revenue function with respect to the selling price. If the marginal revenue is greater than the marginal cost, raising the price will increase profit.
To find the marginal revenue, you need the revenue function, which is given by:
R(p) = p * q
Differentiate the revenue function with respect to the selling price (p):
R'(p) = q
This tells us that the derivative of the revenue function with respect to the price is simply the quantity of goods sold when the selling price is p.
In our case, when the selling price is $9, 36 goods are sold. So, R'(9) = 36.
To determine whether the price should be raised, we need to compare the marginal revenue and the marginal cost. The marginal cost (from part b) is C'(q) = 0.03q^2 - 1.2q + 13.
If the marginal revenue is greater than the marginal cost (R'(9) > C'(q)), then the price should be raised. To find the exact amount by which to raise the price, we need to solve the equation R'(p) = C'(q) for q, where p is the new price we want to find.
Substitute the known values: R'(p) = 36 and C'(q) = 0.03q^2 - 1.2q + 13. Solve the equation 36 = 0.03q^2 - 1.2q + 13 for q to get the quantity of goods produced.
Once you find the value of q, use the revenue function to find the total revenue R(p) when selling q goods at price p. Subtract the total cost C(q) to get the profit, and compare it to the previous profit. The difference will give you the amount by which you should raise the price.
Remember to round the answer to the nearest cent, as specified in the question.