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many hot-water heating systems have a reservoir tank connected directly to the pipeline, so as to allow for expansion when the water becomes hot. the heating system of a house has 52.1 m of copper pipe whose inside radius is 9.68 x 10⁻³ m. when the water and pipe are heated from 23.3 to 70.2 °c, what must be the minimum volume of the reservoir tank to hold the overflow of water?
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Accepted Answer

# Explanation:
## Step1: Calculate the initial volume of water in the pipe
The volume of a cylinder $V=\pi r^{2}L$. Given $r = 9.68\times10^{-3}\text{ m}$ and $L=52.1\text{ m}$.
$$
\begin{align*}
V_{0}&=\pi\times(9.68\times 10^{-3})^{2}\times52.1\
&=\pi\times9.37\times10^{-5}\times52.1\
&\approx1.54\times10^{-2}\text{ m}^{3}
\end{align*}
$$
## Step2: Use the formula for volume expansion
The formula for volume expansion is $\Delta V = V_{0}(\beta_{water}-\beta_{copper})\Delta T$. The coefficient of volume expansion for water $\beta_{water}=207\times 10^{-6}\text{ C}^{-1}$ and for copper $\beta_{copper}=51\times 10^{-6}\text{ C}^{-1}$, and $\Delta T=70.2 - 23.3=46.9\text{ C}$.
$$
\begin{align*}
\Delta V&=V_{0}(\beta_{water}-\beta_{copper})\Delta T\
&=1.54\times 10^{-2}\times(207 - 51)\times10^{-6}\times46.9\
&=1.54\times10^{-2}\times156\times10^{-6}\times46.9\
&=1.54\times156\times46.9\times10^{-8}\
&=1.13\times10^{-3}\text{ m}^{3}
\end{align*}
$$

# Answer:
$1.13\times 10^{-3}\text{ m}^{3}$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the initial volume of water in the pipe

Step2: Use the formula for volume expansion

Answer: