25 multiple choice 1 point solve the problem. the rate of expenditure on a particular machine is given by ( m(x) = 9xsqrt{x^2 + 5} ), where ( x ) is time measured in years. maintenance costs through the second year are $76. find the total maintenance function. ( m(x) = 9(x^2 + 5)^{3/2} + 67 ) ( m(x) = 3(x^2 + 5)^{3/2} + 67 ) ( m(x) = 9(x^2 + 5)^{3/2} - 5 ) ( m(x) = 3(x^2 + 5)^{3/2} - 5 )

Question

Accepted Answer

# Explanation:
## Step1: Integrate the rate function
The rate of expenditure is $ M'(x) = 9x\sqrt{x^2 + 5} $. Let $ u = x^2 + 5 $, then $ du = 2x dx $, so $ \frac{1}{2}du = x dx $.
Integrating $ M'(x) $:
$$
\begin{align*}
M(x)&=\int 9x\sqrt{x^2 + 5}dx\
&=9\int x\sqrt{x^2 + 5}dx\
&=9\cdot\frac{1}{2}\int u^{\frac{1}{2}}du\
&=\frac{9}{2}\cdot\frac{2}{3}u^{\frac{3}{2}} + C\
&= 3(x^2 + 5)^{\frac{3}{2}} + C
\end{align*}
$$
## Step2: Use the initial condition
We know that when $ x = 2 $, $ M(2) = 76 $. Substitute into the function:
$$
\begin{align*}
76&=3((2)^2 + 5)^{\frac{3}{2}} + C\
76&=3(4 + 5)^{\frac{3}{2}} + C\
76&=3(9)^{\frac{3}{2}} + C\
76&=3(27) + C\
76&=81 + C\
C&= 76 - 81\
C&= -5
\end{align*}
$$
So the total maintenance function is $ M(x) = 3(x^2 + 5)^{\frac{3}{2}} - 5 $.
# Answer:
$ M(x) = 3(x^2 + 5)^{\frac{3}{2}} - 5 $ (the last option)

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QUESTION IMAGE

Question

Explanation:

Step1: Integrate the rate function

Step2: Use the initial condition

Answer: