The marginal cost of a product is modeled by
dC/dx =12/Cube root (12x+5)
where x is the number of units. When x = 13, C = 160.
(a) Find the cost function. (Round your constant term to two decimal places.)
C= Ans- 3/2(12x+5)^2/3 +115.61 ????
(b) Find the cost of producing 70 units. (Round your answer to two decimal places.)
$-__???____
dC/dx = 12 (12x+5)^-(1/3)
C = (3/2)(12x+5)^(2/3) + constant agree
160 = (3/2)(156+5)^(2/3) + constant
160 = 44.4 + constant
so
C = (3/2)(12x+5)^(2/3) + 115.61 agree
if x = 70 then
C = 249.68
So what was the problem? The first part was the hard part and you did it with no trouble.
To find the cost function, we need to integrate the marginal cost function with respect to x.
∫(dC/dx) dx = ∫(12/(12x+5)^(1/3)) dx
Let's first simplify the integral:
Let u = 12x + 5.
Then, du/dx = 12, and dx = du/12.
The integral becomes:
∫(12/u^(1/3)) (du/12)
= ∫u^(-1/3) du
= (3u^(2/3))/2 + C
Substitute back u = 12x + 5:
(3(12x + 5)^(2/3))/2 + C
Now, we are given that C = 160 when x = 13. Substitute these values in to find the constant term C:
160 = (3(12*13 + 5)^(2/3))/2 + C
160 = (3(156 + 5)^(2/3))/2 + C
160 = (3(161)^(2/3))/2 + C
320 = 3(161)^(2/3) + 2C
320 - 3(161)^(2/3) = 2C
C = (320 - 3(161)^(2/3))/2
Therefore, the cost function is:
C = (3(12x + 5)^(2/3))/2 + (320 - 3(161)^(2/3))/2
Substituting x = 70 into the cost function will give us the cost of producing 70 units:
C(70) = (3(12*70 + 5)^(2/3))/2 + (320 - 3(161)^(2/3))/2
Calculate this expression to find the cost of producing 70 units.
To find the cost function, we need to integrate the given marginal cost function with respect to x.
The given marginal cost function is dC/dx = 12/cube root(12x + 5).
Integrating the above function will give the cost function C.
∫dC/dx dx = ∫12/cube root(12x + 5) dx
To integrate the expression, we can use a variable substitution. Let u = 12x + 5.
Now, differentiating both sides with respect to x, we get du/dx = 12.
Rearranging the equation, we have dx = du/12.
Substituting the values into the original equation, we get:
∫dC/dx dx = ∫12/cube root(u) du/12
Simplifying the expression, we have:
∫dC = ∫1/cube root(u) du
Now, we can integrate the right side of the equation:
C = 3/2 * u^(2/3) + C
Substituting back the value of u = 12x + 5, we have:
C = 3/2 * (12x + 5)^(2/3) + C
To find the constant term, we can use the given information that when x = 13, C = 160.
Substituting these values into the equation, we have:
160 = 3/2 * (12 * 13 + 5)^(2/3) + C
Solving this equation for C, we find:
C = 3/2 * (157)^(2/3) - 160 ≈ 115.61
Therefore, the cost function is given by:
C = 3/2 * (12x + 5)^(2/3) + 115.61
Now, let's find the cost of producing 70 units by substituting x = 70 into the cost function:
C = 3/2 * (12 * 70 + 5)^(2/3) + 115.61
Calculating the expression, we find:
C ≈ $267.22
Therefore, the cost of producing 70 units is approximately $267.22.