The total cost

C(q)
of producing q goods is given by the following equation.
C(q) = 0.01q^3 − 0.6q^2 + 13q

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.)

I know the fixed cost is 0 but I don't know where else to go from here.

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 find the maximum profit, we need to determine the quantity of goods that will maximize profit. This can be achieved by finding the point where the derivative of the cost function, C(q), is equal to zero.

Given that the cost function is:
C(q) = 0.01q^3 − 0.6q^2 + 13q

To find the derivative of C(q), we take the derivative of each term separately. The derivative of q^n is nq^(n-1).

C'(q) = (0.01*3q^2) - (0.6*2q) + 13
= 0.03q^2 - 1.2q + 13

Setting C'(q) equal to zero, we can solve for q to find the quantity that maximizes profit.

0.03q^2 - 1.2q + 13 = 0

This is a quadratic equation. We can use the quadratic formula to solve for q:

q = (-b ± √(b^2 - 4ac)) / 2a

For this quadratic equation, a = 0.03, b = -1.2, and c = 13.

Using the quadratic formula, we have:
q = (-(-1.2) ± √((-1.2)^2 - 4*0.03*13)) / (2*0.03)
q = (1.2 ± √(1.44 - 1.56)) / 0.06
q = (1.2 ± √(-0.12)) / 0.06

Since the discriminant (√(b^2 - 4ac)) is negative, there are no real solutions for q. This means that the cost function does not have a maximum point in the given range. Therefore, we need to consider a different approach to find the maximum profit.

To determine the optimal quantity and maximum profit, we can consider the revenue function and the profit function.

The revenue function R(q) is given by:
R(q) = price per unit * quantity
= 9q

The profit function P(q) is given by:
P(q) = R(q) - C(q)
= 9q - (0.01q^3 - 0.6q^2 + 13q)
= 9q - 0.01q^3 + 0.6q^2 - 13q

To find the quantity q that maximizes profit, we can take the derivative of the profit function with respect to q and set it equal to zero:

P'(q) = 9 - 0.03q^2 + 1.2q - 13

Setting P'(q) equal to zero:

0.03q^2 - 1.2q + 4 = 0

This is another quadratic equation. Using the quadratic formula, we can solve for q:

q = (-(-1.2) ± √((-1.2)^2 - 4*0.03*4)) / (2*0.03)
q = (1.2 ± √(1.44 - 0.48)) / 0.06
q = (1.2 ± √(0.96)) / 0.06

Simplifying further:

q = (1.2 ± 0.9798) / 0.06

We get two possible solutions:
q ≈ 31.6333 or q ≈ 8.3667

Since q represents the quantity of goods produced, it cannot be in decimal values. Therefore, the optimal quantity for maximizing profit is q = 31.

To find the maximum profit, substitute this value of q back into the profit function:

P(31) = 9(31) - 0.01(31)^3 + 0.6(31)^2 - 13(31)

Calculating this expression will give you the maximum profit value. Round your answer to the nearest cent.

Regarding the second question about whether the price should be raised, we can compare the total revenue generated at the current price ($9) and the projected revenue at a higher price. We should also consider the decrease in quantity sold for each $1 increase in price.

To assess this, we can substitute the value of q = 36 into the profit function P(q) and find the profit at the current price of $9. Then, we can compare it to the projected profit if the price is increased by $1.

To find the profit at the current price, substitute q = 36 into the profit function:
P(36) = 9(36) - 0.01(36)^3 + 0.6(36)^2 - 13(36)

Calculating this expression will give you the current profit.

Next, we can calculate the projected profit if the price is increased by $1. For each $1 increase, 2 fewer goods are sold. Therefore, if the price is increased by $1, the quantity sold would be 36 - (1 * 2) = 34.

Substitute q = 34 into the profit function to find the projected profit with the increased price:
P(34) = 9(34) - 0.01(34)^3 + 0.6(34)^2 - 13(34)

Comparing the current profit and the projected profit, you can determine if raising the price would result in an increase in profit.