I don't understand how to get q for this problem? The next steps for the problem, you just need to plug q in, but I'm not sure what to do?

Do you set revenues equal to expenses?
Problem:
Revenue is given by
R(q) = 750q
and cost is given by
C(q) = 6000 + 5q2.
1) At what quantity is profit maximized?
q =
2) What is the total profit at this production level?

For problem 2, it's clear you just need to plug q in. I just don't know how to get q.
Please help?

You find q by taking the derivatives of both R(q) and C(q) and setting the derivatives equal to each other.

R'(q)=750
C'(q)=10q
q=75
Total profit is $22125, once you plug it in.

profit is revenue - cost

P(q) = R(q)-C(q)
= 750q - (6000+5q^2)
= -5q^2 + 750q - 6000

dP/dq = -10q + 750
So, P is maximuzed when q=75

revenue = cost is the break-even point. Not much profit there...

To find the quantity (q) at which profit is maximized, you need to determine the point at which the difference between revenue and cost is at its highest value. This point is known as the "profit-maximizing quantity."

To find q, start by setting the revenue equation R(q) equal to the cost equation C(q) and solve for q:

750q = 6000 + 5q^2

To simplify this equation, we can move all terms to one side:

5q^2 - 750q + 6000 = 0

Now, you have a quadratic equation. We can use the quadratic formula to solve for q:

q = (-b ± √(b^2 - 4ac)) / 2a

For this equation, a = 5, b = -750, and c = 6000. Plugging in these values, we get:

q = (-(-750) ± √((-750)^2 - 4 * 5 * 6000)) / 2 * 5

Simplifying further:

q = (750 ± √(562500 - 120000)) / 10

q = (750 ± √(442500)) / 10

q = (750 ± 665) / 10

This gives us two potential solutions for q:

1) q = (750 + 665) / 10 = 141.5
2) q = (750 - 665) / 10 = 8.5

It's important to note that since we're dealing with a real-world quantity (in this case, the quantity of a product being produced), we can discard the negative solution (8.5) since it doesn't make sense in this context. Hence, the profit-maximizing quantity is q = 141.5 (rounded to one decimal place).

To find the total profit at this production level (problem 2), simply substitute the value of q = 141.5 into the profit equation P(q) = R(q) - C(q):

P(q) = R(q) - C(q)
= 750q - (6000 + 5q^2)
= 750(141.5) - (6000 + 5(141.5)^2)
= 106125 - (6000 + 10050.25)
= 106125 - 16050.25
= 90074.75

Therefore, the total profit at the production level of q = 141.5 is $90,074.75.