Solve for r v=1/3*(pir^2H)

$v = \frac{1}{3} \cdot (π r^{2} H)$

Step 1

Rewrite the equation as $\frac{1}{3} \cdot (π r^{2} H) = v$ .

$\frac{1}{3} \cdot (π r^{2} H) = v$

Step 2

Multiply both sides of the equation by $\frac{1}{\frac{1}{3} π}$ .

$\frac{1}{\frac{1}{3} π} (\frac{1}{3} \cdot (π r^{2} H)) = \frac{1}{\frac{1}{3} π} v$

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{1}{\frac{1}{3} π} (\frac{1}{3} \cdot (π r^{2} H))$ .

Combine $\frac{1}{3}$ and $π$ .

$\frac{1}{\frac{π}{3}} (\frac{1}{3} \cdot (π r^{2} H)) = \frac{1}{\frac{1}{3} π} v$

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{π} (\frac{1}{3} \cdot (π r^{2} H)) = \frac{1}{\frac{1}{3} π} v$

Multiply $\frac{3}{π}$ by $1$ .

$\frac{3}{π} (\frac{1}{3} \cdot (π r^{2} H)) = \frac{1}{\frac{1}{3} π} v$

Cancel the common factor of $π$ .

Factor $π$ out of $\frac{1}{3} \cdot (π r^{2} H)$ .

$\frac{3}{π} (π (\frac{1}{3} \cdot (r^{2} H))) = \frac{1}{\frac{1}{3} π} v$

Cancel the common factor.

$\frac{3}{π} (π (\frac{1}{3} \cdot (r^{2} H))) = \frac{1}{\frac{1}{3} π} v$

Rewrite the expression.

$3 (\frac{1}{3} \cdot (r^{2} H)) = \frac{1}{\frac{1}{3} π} v$

Combine $r^{2}$ and $\frac{1}{3}$ .

$3 (\frac{r^{2}}{3} \cdot H) = \frac{1}{\frac{1}{3} π} v$

Combine $\frac{r^{2}}{3}$ and $H$ .

$3 \frac{r^{2} H}{3} = \frac{1}{\frac{1}{3} π} v$

Combine $3$ and $\frac{r^{2} H}{3}$ .

$\frac{3 (r^{2} H)}{3} = \frac{1}{\frac{1}{3} π} v$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (r^{2} H)}{3} = \frac{1}{\frac{1}{3} π} v$

Divide $r^{2} H$ by $1$ .

$r^{2} H = \frac{1}{\frac{1}{3} π} v$

Simplify the right side.

Simplify $\frac{1}{\frac{1}{3} π} v$ .

Combine $\frac{1}{3}$ and $π$ .

$r^{2} H = \frac{1}{\frac{π}{3}} v$

Multiply the numerator by the reciprocal of the denominator.

$r^{2} H = 1 \frac{3}{π} v$

Multiply $\frac{3}{π}$ by $1$ .

$r^{2} H = \frac{3}{π} v$

Combine $\frac{3}{π}$ and $v$ .

$r^{2} H = \frac{3 v}{π}$

Step 4

Divide each term in $r^{2} H = \frac{3 v}{π}$ by $H$ and simplify.

Divide each term in $r^{2} H = \frac{3 v}{π}$ by $H$ .

$\frac{r^{2} H}{H} = \frac{\frac{3 v}{π}}{H}$

Simplify the left side.

Cancel the common factor of $H$ .

Cancel the common factor.

$\frac{r^{2} H}{H} = \frac{\frac{3 v}{π}}{H}$

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{\frac{3 v}{π}}{H}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = \frac{3 v}{π} \cdot \frac{1}{H}$

Multiply $\frac{3 v}{π}$ by $\frac{1}{H}$ .

$r^{2} = \frac{3 v}{π H}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\frac{3 v}{π H}}$

Step 6

Simplify $\pm \sqrt{\frac{3 v}{π H}}$ .

Rewrite $\sqrt{\frac{3 v}{π H}}$ as $\frac{\sqrt{3 v}}{\sqrt{π H}}$ .

$r = \pm \frac{\sqrt{3 v}}{\sqrt{π H}}$

Multiply $\frac{\sqrt{3 v}}{\sqrt{π H}}$ by $\frac{\sqrt{π H}}{\sqrt{π H}}$ .

$r = \pm \frac{\sqrt{3 v}}{\sqrt{π H}} \cdot \frac{\sqrt{π H}}{\sqrt{π H}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 v}}{\sqrt{π H}}$ by $\frac{\sqrt{π H}}{\sqrt{π H}}$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{\sqrt{π H} \sqrt{π H}}$

Raise $\sqrt{π H}$ to the power of $1$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{\sqrt{π H}}^{1} \sqrt{π H}}$

Raise $\sqrt{π H}$ to the power of $1$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{\sqrt{π H}}^{1} {\sqrt{π H}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{\sqrt{π H}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{\sqrt{π H}}^{2}}$

Rewrite ${\sqrt{π H}}^{2}$ as $π H$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{π H}$ as ${(π H)}^{\frac{1}{2}}$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{({(π H)}^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{(π H)}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{(π H)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{(π H)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{{(π H)}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{3 v} \sqrt{π H}}{π H}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{3 v π H}}{π H}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{3 v π H}}{π H}$

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{3 v π H}}{π H}$

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{3 v π H}}{π H}$

$r = - \frac{\sqrt{3 v π H}}{π H}$

$r = \frac{\sqrt{3 v π H}}{π H}$

$r = - \frac{\sqrt{3 v π H}}{π H}$

Solve for r v=1/3*(pir^2H)

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