To determine which equations accurately model the relationship where the cube root of \( r \) varies inversely with the square of \( s \), let's analyze how this relationship can be expressed mathematically.
### Step-by-Step Solution:
1. Relationship Statement: The statement "the cube root of \( r \) varies inversely with the square of \( s \)" can be expressed as:
[tex]\[
\sqrt[3]{r} = \frac{k}{s^2}
\][/tex]
where \( k \) is a constant.
2. Rewrite Relationship:
- Begin with the given expression:
[tex]\[
\sqrt[3]{r} = \frac{k}{s^2}
\][/tex]
- Multiplying both sides by \( s^2 \) to remove the fraction:
[tex]\[
s^2 \sqrt[3]{r} = k
\][/tex]
3. Examine Options:
- Option 1: \(\sqrt[3]{r} = \frac{k}{s^2}\)
- This directly matches our derived equation.
- Correct.
- Option 2: \(s^{\frac{1}{2}} r^3 = k\)
- Not compatible with the inverse cube root and square relationship.
- Incorrect.
- Option 3: \(\frac{s^2}{\sqrt[3]{r}} = k\)
- In radians, this does not match any form we derived.
- Incorrect.
- Option 4: \(\sqrt[3]{r} = s^2 k\)
- Inconsistent with the original inverse relationship.
- Incorrect.
- Option 5: \(s^2 r^{\frac{1}{3}} = k\)
- This is simply another form of our modified equation \(s^2 \sqrt[3]{r} = k\).
- Correct.
- Option 6: \(\frac{\sqrt[3]{r}}{s^3} = k\)
- This does not adhere to the original inverse relationship.
- Incorrect.
### Correct Equations:
Based on the above evaluations, the two equations that accurately model the given relationship are:
[tex]\[
\sqrt[3]{r} = \frac{k}{s^2}
\][/tex]
and
[tex]\[
s^2 r^{\frac{1}{3}} = k
\][/tex]
Thus, the correct answers are:
Option 1 and Option 5.
