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.
the total cost is the integral of the marginal cost. So, it is
∫[0,10] (q^2-10q+30) dq = 400/3
To find the total cost of manufacturing 10 boxes, we need to integrate the marginal cost function c'(q) to get the total cost function c(q) and then evaluate it at q = 10.
The marginal cost function, c'(q), is given as q^2 - 10q + 30.
To find the total cost function, we need to integrate c'(q) with respect to q. Let's integrate each term of the function:
∫(q^2 - 10q + 30) dq
= ∫q^2 dq - ∫10q dq + ∫30 dq
Now, applying the power rule of integration, we get:
= (1/3)q^3 - (10/2)q^2 + 30q + C
Where C is the constant of integration.
Now let's substitute q = 10 into the total cost function c(q):
c(10) = (1/3)(10)^3 - (10/2)(10)^2 + 30(10) + C
= (1/3)(1000) - (10/2)(100) + 300 + C
= 333.33 - 500 + 300 + C
= 133.33 + C
Since we are rounding to the nearest dollar, we can simply round 133.33 to 133.
Therefore, the total cost of manufacturing 10 boxes is approximately $133.