The marginal cost of a product is modeled by
dC/dx= 16/cube root 16x + 3
where x is the number of units. When x = 17, C = 120.
(a) Find the cost function. (Round your constant term to two decimal places.)
c=
(b) Find the cost of producing 90 units. (Round your answer to two decimal places.)
$
To find the cost function, we need to integrate the marginal cost function with respect to x:
∫(dC/dx) dx = ∫(16/(16x)^(1/3) + 3) dx
Using the power rule of integration, we have:
∫(dC/dx) dx = ∫(16/(16x)^(1/3)) dx + ∫(3) dx
= 16∫(16x)^(-1/3) dx + 3∫(1) dx
= 16 * (3/2) * (16x)^(2/3) + 3x + C
Simplifying this expression, we get:
C = 24 * (16x)^(2/3) + 3x + C_1
Now, we can use the given information when x = 17, C = 120 to solve for the constant term C_1.
120 = 24 * (16*17)^(2/3) + 3*17 + C_1
120 = 24 * 16^(2/3) * 17^(2/3) + 51 + C_1
Now, we can solve for C_1:
C_1 = 120 - 24 * 16^(2/3) * 17^(2/3) - 51
C_1 ≈ -7.67 (rounded to two decimal places)
(a) The cost function, c, is given by:
c = 24 * (16x)^(2/3) + 3x - 7.67
(b) To find the cost of producing 90 units, we substitute x = 90 into the cost function:
c = 24 * (16*90)^(2/3) + 3*90 - 7.67
c ≈ $6,215.81 (rounded to two decimal places)
Therefore, the cost of producing 90 units is approximately $6,215.81.