Carlota Music Company estimates that the marginal cost of manufacturing its Professional Services guitars in dollars/month by the following. This model assumes that the level of production is x guitars/month.
C '(x) = 0.007x + 130
The fixed costs incurred by Carlota are $9500/month. Find the total monthly cost incurred by Carlota in manufacturing x guitars/month.
C(x) =
To find the total monthly cost incurred by Carlota in manufacturing x guitars/month, we need to integrate the marginal cost function C'(x) and then add the fixed costs.
The marginal cost function C'(x) represents the rate of change of the total cost with respect to the number of guitars produced. By integrating this function, we can find the total cost function C(x).
To integrate C'(x), we treat it as a polynomial function, where the coefficient of x is 0.007 and the constant term is 130. We add 1 to the exponent of x and divide the coefficient of x by the new exponent. The integration of C'(x) is as follows:
∫C'(x) dx = 0.007x^2/2 + 130x + C
In this equation, C is the constant of integration. Since we are interested in the total monthly cost, we need to find the particular solution that satisfies the fixed costs. We know that the fixed costs incurred by Carlota are $9500/month, so when x = 0, C(x) should equal 9500.
Substituting x = 0 and C(x) = 9500 into the integrated equation, we can solve for the constant of integration C:
9500 = 0.007(0)^2/2 + 130(0) + C
9500 = 0 + 0 + C
C = 9500
Now that we have determined the constant of integration C, we can write the total cost function C(x) as follows:
C(x) = 0.007x^2/2 + 130x + 9500
Therefore, the total monthly cost incurred by Carlota in manufacturing x guitars/month is given by the function C(x) = 0.007x^2/2 + 130x + 9500.
C(x)=(0.007/2)x^2 + 130x
C(x) =0.0035x^2 +130x +9500
To find the total monthly cost incurred by Carlota in manufacturing x guitars/month, we need to integrate the marginal cost function and add the fixed costs.
First, let's integrate the marginal cost function C '(x) = 0.007x + 130 with respect to x:
∫ (0.007x + 130) dx = 0.007∫ x dx + ∫ 130 dx
= 0.007(0.5x^2) + 130x + C
= 0.0035x^2 + 130x + C1
Now, let's add the fixed costs incurred by Carlota:
C(x) = 0.0035x^2 + 130x + C1 + 9500
The total monthly cost incurred by Carlota in manufacturing x guitars/month is given by the function C(x) = 0.0035x^2 + 130x + C1 + 9500, where C1 is the constant of integration.
To find the total monthly cost incurred by Carlota in manufacturing x guitars/month, we need to calculate the sum of the fixed costs and the variable costs.
Variable costs can be calculated by multiplying the number of guitars produced (x) by the marginal cost per guitar, given by C '(x).
C(x) = Fixed Costs + Variable Costs
Variable Costs = C '(x) * x
Substituting the values into the equation:
C(x) = 9500 + (0.007x + 130) * x
Now, we can simplify and calculate the total monthly cost incurred by Carlota in manufacturing x guitars/month.