Precalculus Examples

$\frac{2}{3} \cdot (p r^{3}) + 30 p r^{2} - 15000 = 0$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{2}{3} (p r^{3})$ .

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Step 1.1.1

Combine $p$ and $\frac{2}{3}$ .

$\frac{p \cdot 2}{3} r^{3} + 30 p r^{2} - 15000 = 0$

Step 1.1.2

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

$\frac{p \cdot 2 r^{3}}{3} + 30 p r^{2} - 15000 = 0$

Step 1.2

Move $2$ to the left of $p$ .

$\frac{2 p r^{3}}{3} + 30 p r^{2} - 15000 = 0$

Step 2

Add $15000$ to both sides of the equation.

$\frac{2 p r^{3}}{3} + 30 p r^{2} = 15000$

Step 3

Factor $2 p r^{2}$ out of $\frac{2 p r^{3}}{3} + 30 p r^{2}$ .

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Step 3.1

Factor $2 p r^{2}$ out of $\frac{2 p r^{3}}{3}$ .

$2 p r^{2} (\frac{r}{3}) + 30 p r^{2} = 15000$

Step 3.2

Factor $2 p r^{2}$ out of $30 p r^{2}$ .

$2 p r^{2} (\frac{r}{3}) + 2 p r^{2} (15) = 15000$

Step 3.3

Factor $2 p r^{2}$ out of $2 p r^{2} \frac{r}{3} + 2 p r^{2} (15)$ .

$2 p r^{2} (\frac{r}{3} + 15) = 15000$

Step 4

Divide each term in $2 p r^{2} (\frac{r}{3} + 15) = 15000$ by $2 r^{2} (\frac{r}{3} + 15)$ and simplify.

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Step 4.1

Divide each term in $2 p r^{2} (\frac{r}{3} + 15) = 15000$ by $2 r^{2} (\frac{r}{3} + 15)$ .

$\frac{2 p r^{2} (\frac{r}{3} + 15)}{2 r^{2} (\frac{r}{3} + 15)} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 p r^{2} (\frac{r}{3} + 15)}{2 r^{2} (\frac{r}{3} + 15)} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{p r^{2} (\frac{r}{3} + 15)}{r^{2} (\frac{r}{3} + 15)} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2.2

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

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Step 4.2.2.1

Cancel the common factor.

$\frac{p r^{2} (\frac{r}{3} + 15)}{r^{2} (\frac{r}{3} + 15)} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2.2.2

Rewrite the expression.

$\frac{p (\frac{r}{3} + 15)}{\frac{r}{3} + 15} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2.3

Cancel the common factor of $\frac{r}{3} + 15$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{p (\frac{r}{3} + 15)}{\frac{r}{3} + 15} = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.2.3.2

Divide $p$ by $1$ .

$p = \frac{15000}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $15000$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $15000$ .

$p = \frac{2 \cdot 7500}{2 r^{2} (\frac{r}{3} + 15)}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 r^{2} (\frac{r}{3} + 15)$ .

$p = \frac{2 \cdot 7500}{2 (r^{2} (\frac{r}{3} + 15))}$

Step 4.3.1.2.2

Cancel the common factor.

$p = \frac{2 \cdot 7500}{2 (r^{2} (\frac{r}{3} + 15))}$

Step 4.3.1.2.3

Rewrite the expression.

$p = \frac{7500}{r^{2} (\frac{r}{3} + 15)}$

Step 4.3.2

Simplify the denominator.

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Step 4.3.2.1

To write $15$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$p = \frac{7500}{r^{2} (\frac{r}{3} + 15 \cdot \frac{3}{3})}$

Step 4.3.2.2

Combine $15$ and $\frac{3}{3}$ .

$p = \frac{7500}{r^{2} (\frac{r}{3} + \frac{15 \cdot 3}{3})}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$p = \frac{7500}{r^{2} \frac{r + 15 \cdot 3}{3}}$

Step 4.3.2.4

Multiply $15$ by $3$ .

$p = \frac{7500}{r^{2} \frac{r + 45}{3}}$

Step 4.3.3

Combine $r^{2}$ and $\frac{r + 45}{3}$ .

$p = \frac{7500}{\frac{r^{2} (r + 45)}{3}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$p = 7500 \frac{3}{r^{2} (r + 45)}$

Step 4.3.5

Multiply $7500 \frac{3}{r^{2} (r + 45)}$ .

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Step 4.3.5.1

Combine $7500$ and $\frac{3}{r^{2} (r + 45)}$ .

$p = \frac{7500 \cdot 3}{r^{2} (r + 45)}$

Step 4.3.5.2

Multiply $7500$ by $3$ .

$p = \frac{22500}{r^{2} (r + 45)}$

Step 5

To rewrite as a function of $r$ , write the equation so that $p$ is by itself on one side of the equal sign and an expression involving only $r$ is on the other side.

$f (r) = \frac{22500}{r^{2} (r + 45)}$