The marginal cost is given by C'(x)=x^(1/3)+9. If the fixed costs are $175, find the cost of producing 8 units.
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Take the integral. The initial cost / fixed value (the omni-present +C) is 175. This equation is C(x).
You want to know the cost of producing 8 units, so then calculate C(8).
To find the cost of producing 8 units, we need to calculate the total cost, which includes both the fixed costs and the variable costs.
First, we find the variable cost by integrating the marginal cost function C'(x). Since C'(x) is the derivative of the cost function C(x), integrating C'(x) will give us the cost function:
C(x) = ∫[C'(x) dx]
Given C'(x) = x^(1/3) + 9, let's integrate it to get C(x):
C(x) = ∫[x^(1/3) + 9 dx]
= (3/4)x^(4/3) + 9x + C1
Next, we need to determine the value of the constant C1. Since we have the fixed cost, we can find C1 by setting x to 0 in the cost function:
C(0) = (3/4)(0)^(4/3) + 9(0) + C1
= 0 + 0 + C1
= C1
Therefore, C1 = $175 (the fixed costs provided).
Now, we have our complete cost function:
C(x) = (3/4)x^(4/3) + 9x + 175
To find the cost of producing 8 units, substitute x = 8 into the cost function:
C(8) = (3/4)(8)^(4/3) + 9(8) + 175
= (3/4)(4^3) + 72 + 175
= 3(4) + 72 + 175
= 12 + 72 + 175
= 259 + 175
= $434
Therefore, the cost of producing 8 units is $434.