If \( x \) varies inversely with the cube of \( y \), we can express this relationship mathematically as:
\[ x = \frac{k}{y^3} \]
where \( k \) is a constant.
Given that \( x = 10 \) when \( y = 10 \), we can substitute these values into the equation to find \( k \):
\[ 10 = \frac{k}{10^3} \implies 10 = \frac{k}{1000} \]
Multiplying both sides by 1000:
\[ k = 10 \times 1000 = 10000 \]
Now we have the equation:
\[ x = \frac{10000}{y^3} \]
Next, we need to find \( y \) when \( x = 80 \):
\[ 80 = \frac{10000}{y^3} \]
To solve for \( y^3 \), we first multiply both sides by \( y^3 \):
\[ 80y^3 = 10000 \]
Now divide both sides by 80:
\[ y^3 = \frac{10000}{80} = 125 \]
Taking the cube root of both sides gives us:
\[ y = \sqrt[3]{125} = 5 \]
Thus, when \( x = 80 \), the corresponding value of \( y \) is:
\[ \boxed{5} \]