Calculus Examples

$v = \frac{1}{3} π h^{2} (3 r - h)$

Step 1

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

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

Step 2

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

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

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Combine $\frac{1}{3}$ and $π$ .

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

Step 3.1.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.1.3

Multiply $\frac{3}{π}$ by $1$ .

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

Step 3.1.1.4

Cancel the common factor of $π$ .

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Step 3.1.1.4.1

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

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

Step 3.1.1.4.2

Cancel the common factor.

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

Step 3.1.1.4.3

Rewrite the expression.

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

Step 3.1.1.5

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

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

Step 3.1.1.6

Combine $\frac{h^{2}}{3}$ and $3$ .

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

Step 3.1.1.7

Cancel the common factor of $3$ .

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Step 3.1.1.7.1

Cancel the common factor.

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

Step 3.1.1.7.2

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

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

Step 3.1.1.8

Apply the distributive property.

$h^{2} (3 r) + h^{2} (- h) = \frac{1}{\frac{1}{3} π} v$

Step 3.1.1.9

Reorder.

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Step 3.1.1.9.1

Rewrite using the commutative property of multiplication.

$3 h^{2} r + h^{2} (- h) = \frac{1}{\frac{1}{3} π} v$

Step 3.1.1.9.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.1.10

Multiply $h^{2}$ by $h$ by adding the exponents.

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Step 3.1.1.10.1

Move $h$ .

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

Step 3.1.1.10.2

Multiply $h$ by $h^{2}$ .

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Step 3.1.1.10.2.1

Raise $h$ to the power of $1$ .

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

Step 3.1.1.10.2.2

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

$3 h^{2} r - h^{1 + 2} = \frac{1}{\frac{1}{3} π} v$

Step 3.1.1.10.3

Add $1$ and $2$ .

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

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Combine $\frac{1}{3}$ and $π$ .

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

Step 3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$3 h^{2} r - h^{3} = 1 \frac{3}{π} v$

Step 3.2.1.3

Multiply $\frac{3}{π}$ by $1$ .

$3 h^{2} r - h^{3} = \frac{3}{π} v$

Step 3.2.1.4

Combine $\frac{3}{π}$ and $v$ .

$3 h^{2} r - h^{3} = \frac{3 v}{π}$

Step 4

Add $h^{3}$ to both sides of the equation.

$3 h^{2} r = \frac{3 v}{π} + h^{3}$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Cancel the common factor of $h^{2}$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Divide $r$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.1.2

Combine.

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

Step 5.3.1.3

Cancel the common factor of $3$ .

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Step 5.3.1.3.1

Cancel the common factor.

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

Step 5.3.1.3.2

Rewrite the expression.

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

Step 5.3.1.4

Multiply $v$ by $1$ .

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

Step 5.3.1.5

Cancel the common factor of $h^{3}$ and $h^{2}$ .

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Step 5.3.1.5.1

Factor $h^{2}$ out of $h^{3}$ .

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

Step 5.3.1.5.2

Cancel the common factors.

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Step 5.3.1.5.2.1

Factor $h^{2}$ out of $3 h^{2}$ .

$r = \frac{v}{π h^{2}} + \frac{h^{2} h}{h^{2} \cdot 3}$

Step 5.3.1.5.2.2

Cancel the common factor.

$r = \frac{v}{π h^{2}} + \frac{h^{2} h}{h^{2} \cdot 3}$

Step 5.3.1.5.2.3

Rewrite the expression.

$r = \frac{v}{π h^{2}} + \frac{h}{3}$