the marginal cost function for a manufacturing company is given by dollars per box,c'(q)=q^2-10q+30 where q is the number of boxes manufactured. If , find the total cost of manufacturing 10 boxes. Round to the nearest dollar.
dvd
To find the total cost of manufacturing 10 boxes, we need to integrate the marginal cost function from 0 to 10. The integral of the marginal cost function will give us the total cost function.
The marginal cost function is given as c'(q) = q^2 - 10q + 30. To find the total cost function c(q), we integrate c'(q):
c(q) = ∫(q^2 - 10q + 30) dq
Integrated, this becomes:
c(q) = (1/3)q^3 - 5q^2 + 30q + C
where C is the constant of integration.
Now, we need to find C. Since C represents the cost when q = 0, we can use the given information that the company manufactures 10 boxes. So, we substitute q = 10 into the total cost function and solve for C:
c(10) = (1/3)(10)^3 - 5(10)^2 + 30(10) + C
c(10) = 1000/3 - 500 + 300 + C
c(10) = 800/3 + C
We know that the total cost for manufacturing 10 boxes is given as $1680. So, we set c(10) equal to 1680 and solve for C:
800/3 + C = 1680
C = 1680 - 800/3
C = 5040/3 - 800/3
C = 4240/3
Now that we have the constant of integration, we can substitute it back into the total cost function:
c(q) = (1/3)q^3 - 5q^2 + 30q + 4240/3
To find the total cost of manufacturing 10 boxes, we substitute q = 10 into the total cost function:
c(10) = (1/3)(10)^3 - 5(10)^2 + 30(10) + 4240/3
Evaluating this expression gives us the total cost of manufacturing 10 boxes.