#### Answer

a) $450$ houses can be served by a water pipe with a 30-centimeter diameter.
b) A water pipe with $25\text{ centimeter}$ diameter can serve a new subdivision of 1250 houses.

#### Work Step by Step

(a)
We have,
$\begin{align}
& h\propto {{d}^{2}} \\
& h=k{{d}^{2}} \\
\end{align}$
Here, h is the number of houses served and d is the diameter of the pipe.
Substitute the value of d and h to get the value of k.
$\begin{align}
& h=k{{d}^{2}} \\
& 50=k\cdot {{(10)}^{2}} \\
& k=\frac{50}{100} \\
& k=0.5
\end{align}$
Substitute $d=30$ and $k=0.5$in $h=k{{d}^{2}}$.
$\begin{align}
& h=0.5{{d}^{2}} \\
& h=0.5{{(30)}^{2}} \\
& h=450 \\
\end{align}$
Therefore, 450 houses can be served by the water pipe with a 30-centimeter diameter.
(b)
Substitute $h=50$ and $d=10$ in $h=k{{d}^{2}}$
$\begin{align}
& h=k{{d}^{2}} \\
& 50=k\cdot {{(10)}^{2}} \\
& k=\frac{50}{100} \\
& k=0.5
\end{align}$
As,$h=0.5{{d}^{2}}$
For $h=1250$ ,
$\begin{align}
& h=0.5{{d}^{2}} \\
& 1250=0.5{{d}^{2}} \\
& {{d}^{2}}=\frac{1250}{0.5} \\
& {{d}^{2}}=625 \\
& d=\sqrt{625} \\
& d=25 \\
\end{align}$
A water pipe with a 25-centimeter diameter can serve a new subdivision of 1250 houses.