Question

1. solve for r (radius) in the formula for the volume of a sphere: $v = \frac{4}{3}\pi r^{3}$
2. solve for s (side length) in the formula for the volume of a cube: $v = s^{3}$
3. solve for r (radius) in the formula for the surface area of a sphere: $a = 4\pi r^{2}$
4. solve for h (height) in the formula for the volume of a cone: $v = \frac{1}{3}\pi r^{2}h$
5. solve for x (side length) in the formula for the diagonal of a cube: $d = x\sqrt{3}$

Explanation:

Step1: Isolate $r^3$

Starting with $V = \frac{4}{3}\pi r^{3}$, multiply both sides by $\frac{3}{4\pi}$ to get $r^{3}=\frac{3V}{4\pi}$.

Step2: Solve for $r$

Take the cube - root of both sides. So, $r=\sqrt[3]{\frac{3V}{4\pi}}$.

Step1: Solve for $s$

Given $V = s^{3}$, take the cube - root of both sides. So, $s=\sqrt[3]{V}$.

Step1: Isolate $r^{2}$

Starting with $A = 4\pi r^{2}$, divide both sides by $4\pi$ to get $r^{2}=\frac{A}{4\pi}$.

Step2: Solve for $r$

Take the square - root of both sides. Since $r$ represents a radius (a non - negative quantity), $r=\sqrt{\frac{A}{4\pi}}=\frac{1}{2}\sqrt{\frac{A}{\pi}}$.

Step1: Isolate $h$

Starting with $V=\frac{1}{3}\pi r^{2}h$, multiply both sides by $3$ to get $3V=\pi r^{2}h$. Then divide both sides by $\pi r^{2}$ to obtain $h=\frac{3V}{\pi r^{2}}$.

Answer:

$r=\sqrt[3]{\frac{3V}{4\pi}}$