Question

solve the formula for the indicated variable.

1. volume of a cone: $v=\frac{1}{3}pi r^{2}h$; solve for $h$.
2. perimeter of a rectangle: $p = 2ell+2w$; solve for $w$.
3. area of a rectangle: $a=ell w$; solve for $ell$.
4. the surface area of a right - circular cylinder is given by the formula $s = 2pi rh+2pi r^{2}$. solve the equation for $h$.

Explanation:

Step1: Isolate $h$ in the cone - volume formula

Given $V=\frac{1}{3}\pi r^{2}h$. Multiply both sides by 3 to get $3V = \pi r^{2}h$. Then divide both sides by $\pi r^{2}$ to solve for $h$. So $h=\frac{3V}{\pi r^{2}}$.

Step2: Isolate $w$ in the rectangle - perimeter formula

Given $P = 2\ell+2w$. First, subtract $2\ell$ from both sides: $P - 2\ell=2w$. Then divide both sides by 2: $w=\frac{P - 2\ell}{2}$.

Step3: Isolate $\ell$ in the rectangle - area formula

Given $A=\ell w$. Divide both sides by $w$ (assuming $w
eq0$) to get $\ell=\frac{A}{w}$.

Step4: Isolate $h$ in the cylinder - surface - area formula

Given $S = 2\pi rh+2\pi r^{2}$. First, subtract $2\pi r^{2}$ from both sides: $S - 2\pi r^{2}=2\pi rh$. Then divide both sides by $2\pi r$ (assuming $r
eq0$) to get $h=\frac{S - 2\pi r^{2}}{2\pi r}$.

Answer:

1. $h=\frac{3V}{\pi r^{2}}$
2. $w=\frac{P - 2\ell}{2}$
3. $\ell=\frac{A}{w}$
4. $h=\frac{S - 2\pi r^{2}}{2\pi r}$