To find the cost function when given the marginal cost function [tex]\( C'(x) = x^{1/4} + 6 \)[/tex] and the information that 16 units cost [tex]$132, we'll follow these steps:
1. Integrate the Marginal Cost Function:
The cost function \( C(x) \) is the integral of the marginal cost function \( C'(x) \). So, we need to integrate \( x^{1/4} + 6 \).
\[
C(x) = \int (x^{1/4} + 6) \, dx
\]
2. Compute the Indefinite Integral:
Integrate \( x^{1/4} \) and \( 6 \) separately.
\[
\int x^{1/4} \, dx
\]
Recall that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \).
\[
\int x^{1/4} \, dx = \frac{x^{1/4 + 1}}{1/4 + 1} = \frac{x^{5/4}}{5/4} = \frac{4}{5} x^{5/4}
\]
Now, integrate the constant 6:
\[
\int 6 \, dx = 6x
\]
Combining these, we get the indefinite integral of \( C'(x) \):
\[
C(x) = \frac{4}{5} x^{5/4} + 6x + C
\]
Here, \( C \) represents the constant of integration.
3. Determine the Constant of Integration \( C \):
Use the given condition that 16 units cost $[/tex]132. This means [tex]\( C(16) = 132 \)[/tex].
Substitute [tex]\( x = 16 \)[/tex] and [tex]\( C(16) = 132 \)[/tex]:
[tex]\[
132 = \frac{4}{5} (16)^{5/4} + 6(16) + C
\][/tex]
Calculate [tex]\( (16)^{5/4} \)[/tex]:
[tex]\[
16 = 2^4 \implies 16^{5/4} = (2^4)^{5/4} = 2^5 = 32
\][/tex]
Substitute [tex]\( 16^{5/4} = 32 \)[/tex] into the equation:
[tex]\[
132 = \frac{4}{5} \cdot 32 + 96 + C
\][/tex]
Simplify:
[tex]\[
132 = \frac{128}{5} + 96 + C
\][/tex]
Combine the terms:
[tex]\[
132 = 25.6 + 96 + C
\][/tex]
[tex]\[
132 = 121.6 + C
\][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[
C = 132 - 121.6 = 10.4
\][/tex]
4. Write the Final Cost Function:
Substitute [tex]\( C = 10.4 \)[/tex] back into the cost function:
[tex]\[
C(x) = \frac{4}{5} x^{5/4} + 6x + 10.4
\][/tex]
Thus, the cost function is:
[tex]\[
C(x) = \frac{4}{5} x^{5/4} + 6x + 10.4
\][/tex]
