Carla Music Company estimates that the marginal cost of manufacturing its professional series guitar is C’ (x) = .002x + 100 dollars/month when the level of production is x guitars/month. The fixed costs by Carla are $4000 per month.
A. Find the total monthly cost incurred by Carla in manufacturing x guitars per month.
B. What is the cost of producing 50 guitars per month.
I am soo confused on this problem and I have no idea how to even start. can someone please help me?!?
If you typed it correctly, your equation is the derivative of C, so integrate it
C = .001x^2 + 100x + 4000
B:
C = .001(2500) + 100(50) + 4000
=
=.....
Sure! I can help you with this problem. Let's break it down step by step.
A. To find the total monthly cost incurred by Carla in manufacturing x guitars per month, we need to consider both the fixed costs and the marginal costs.
1. Fixed costs: Carla's fixed costs are $4000 per month. This cost does not depend on the number of guitars produced.
2. Marginal costs: The marginal cost function is given as C'(x) = 0.002x + 100 dollars per month. This represents the additional cost incurred for each additional guitar produced.
To find the total marginal cost, we need to integrate the marginal cost function. Integrating the function will give us the cumulative or total cost function.
∫(0.002x + 100) dx = 0.001x^2 + 100x + C1
Here C1 is the constant of integration.
Now we have the total cost function, which includes both fixed costs and marginal costs:
C(x) = 0.001x^2 + 100x + 4000
B. To find the cost of producing 50 guitars per month, we can substitute x = 50 into the total cost function:
C(50) = 0.001(50)^2 + 100(50) + 4000
= 2.5 + 500 + 4000
= 5002.5 dollars
Therefore, the cost of producing 50 guitars per month is $5002.5.
So, the total monthly cost incurred by Carla in manufacturing x guitars per month is given by the function C(x) = 0.001x^2 + 100x + 4000, and the cost of producing 50 guitars per month is $5002.5.