Use the given information to find the unknown value.
$y$ varies inversely with the cube of $x$. When $x=5$, then $y=1$. Find $y$ when $x=1$
Enter the exact answer.
\[
y=\text { Number }
\]
Answer
\(\boxed{y = 125}\) when $x = 1$.
Step 1 ：Given that $y$ varies inversely with the cube of $x$, we can write this relationship as: \(y * x^3 = k\), where k is the constant of variation.
Step 2 ：Using the given information, when $x=5$, $y=1$. We can plug these values into the equation to find the value of k: \(1 * 5^3 = k\) which gives us \(k = 125\).
Step 3 ：Now, we need to find the value of $y$ when $x=1$. We can rewrite the equation as: \(y = \frac{k}{x^3}\).
Step 4 ：Plugging in the values of $k$ and $x$, we get: \(y = \frac{125}{1^3}\), which simplifies to \(y = 125\).
Step 5 ：\(\boxed{y = 125}\) when $x = 1$.
