Question

select all the correct answers.

the cube root of \\(r\\) varies inversely with the square of \\(s\\). which two equations model this relationship?

\\(\square\\ \frac{\sqrt3{r}}{s^3} = k\\)

\\(\square\\ s^2 r^{\frac{1}{3}} = k\\)

\\(\square\\ \sqrt3{r} = s^2 k\\)

\\(\square\\ s^{\frac{1}{2}} r^3 = k\\)

\\(\square\\ \frac{s^2}{\sqrt3{r}} = k\\)

\\(\square\\ \sqrt3{r} = \frac{k}{s^2}\\)

Explanation:

Translate the verbal relationship into an equation

Using the Inverse Variation knowledge point

$$ \sqrt[3]{r} = \frac{k}{s^2} $$

Rewrite the radical expression using rational exponents

Using the Rational Exponents knowledge point

$$ \sqrt[3]{r} = r^{\frac{1}{3}} $$

Rearrange the equation to find equivalent forms

Using the Inverse Variation knowledge point

$$
LATEXBLOCK0
$$

Identify the matching options from the given list

Using the Inverse Variation knowledge point

$$
LATEXBLOCK1
$$

Answer:

• (A) \(\frac{\sqrt[3]{r}}{s^3} = k\)
• (B) \(s^2 r^{\frac{1}{3}} = k\) (Correct answer)
• (C) \(\sqrt[3]{r} = s^2 k\)
• (D) \(s^{\frac{1}{2}} r^3 = k\)
• (E) \(\frac{s^2}{\sqrt[3]{r}} = k\)
• (F) \(\sqrt[3]{r} = \frac{k}{s^2}\) (Correct answer)