Question

Suppose x varies inversely with the cube of y. If x=10
when y=10
, what is y when x is 80?

Answer 1

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:

\[
\boxed{5}
\]