To understand which equations model the relationship "the cube root of \( r \) varies inversely with the square of \( s \)," we need to break down the statement mathematically. Specifically, we need to identify the equations where \(\sqrt[3]{r}\) is expressed as some constant divided by \(s^2\).
Let's evaluate each equation step by step:
1. \(\sqrt[3]{r} = \frac{k}{s^2}\)
This equation directly states that the cube root of \( r \) is equal to a constant \( k \) divided by the square of \( s \). This matches the given relationship precisely. So this equation is correct.
2. \(s^{\frac{1}{2}} r ^3 = k\)
Here, \( s^{\frac{1}{2}} r^3 \) is equal to a constant \( k \). This does not directly match the form \(\sqrt[3]{r} = \frac{k}{s^2}\), so this equation does not model the given relationship.
3. \(\frac{s^2}{\sqrt[3]{r}} = k\)
This can be rearranged to \(\sqrt[3]{r} = \frac{s^2}{k} \). Here, we see that \(\sqrt[3]{r}\) is the constant \( k \) divided by \( s^2 \), matching the given relationship. So this equation is correct.
4. \(\sqrt[3]{r} = s^2 k\)
In this equation, \(\sqrt[3]{r}\) is equal to \( k \) multiplied by \( s^2 \), not divided by \( s^2\). This does not match the given relationship. So this equation is not correct.
5. \(s^2 r^{\frac{1}{3}} = k\)
Rearranging this, we get \(r^{\frac{1}{3}} = \frac{k}{s^2}\), which aligns perfectly with the given relationship. So this equation is correct.
6. \(\frac{\sqrt[3]{r}}{s^3} = k\)
This equation can be rearranged to \(\sqrt[3]{r} = k s^3\), which indicates that \(\sqrt[3]{r}\) varies directly, not inversely, with \( s^3 \). So this equation is not correct.
Therefore, the two equations that correctly model the relationship where the cube root of \( r \) varies inversely with the square of \( s \) are:
[tex]\[
\boxed{\sqrt[3]{r} = \frac{k}{s^2} \quad \text{and} \quad \frac{s^2}{\sqrt[3]{r}} = k}
\][/tex]
or, in the list format:
1. \(\sqrt[3]{r} = \frac{k}{s^2}\)
3. \(\frac{s^2}{\sqrt[3]{r}} = k\)
These are the correct answers.
