Question

Cost of Producing Guitars Carlota Music Company estimates that the marginal cost of manufacturing its Professional Series guitars is given by the following in dollars/month when the level of production is x guitars/month.
C '(x) = 0.008x + 90

The fixed costs incurred by Carlota are $8500/month. Find the total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month.

Answer 1

Answer:

The total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month is C(x) = 0.004x^2 + 90x + 8,500.

Explanation:

Given,

C '(x) = 0.008x + 90 ................................... (1)

To obtain the the total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month, we obtain the integral of equation (1) as follows:

$C(x)=\int\limits {C'(x)} \, dx = \int\limits {[0.008x + 90]} \, dx$

C(x) = (0.008 / 2) x^2 + 90x + F

C(x) = 0.004x^2 + 90x + F .......................... (2)

Where F is the constant.

Since total cost is the addition of the total cost and total variable cost, the F in equation (2) represents the total fixed cost per month.

Since the fixed costs incurred by Carlota are $8500/month, this implies that F = 8,500.

Substituting F = 8,500 into equation (2), we have:

C(x) = 0.004x^2 + 90x + 8,500 <-------------- Total cost per month

Therefore, the total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month is C(x) = 0.004x^2 + 90x + 8,500.