$y$ is directly proportional to $x^{2}$. Given that $y=112.5$ when $x=5$, work out the value of $y$ when $x=8$. If your answer is a decimal, then round it to 1 d.p.

Step 1 ：Given that $y$ is directly proportional to $x^{2}$, we can write the equation as $y=kx^{2}$, where $k$ is the constant of proportionality.

Step 2 ：Using the given values of $x=5$ and $y=112.5$, we can find the value of $k$. Substitute these values into the equation: $112.5=k(5)^{2}$

Step 3 ：Solve for $k$: $k=\frac{112.5}{(5)^{2}}=\frac{112.5}{25}=4.5$

Step 4 ：Now we have the constant of proportionality, $k=4.5$. We can use this to find the value of $y$ when $x=8$. Substitute these values into the equation: $y=4.5(8)^{2}$

Step 5 ：Calculate the value of $y$: $y=4.5(64)=288.0$

Step 6 ：\(\boxed{288.0}\) is the value of $y$ when $x=8$.