Calculus Examples

$- \frac{12000}{r^{2}} + \frac{16 π r}{3} = 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r^{2}, 3, 1$

Step 1.2

Since $r^{2}, 3, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 3, 1$ then find LCM for the variable part $r^{2}$ .

Step 1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 1.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.7

The LCM of $1, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 1.8

The factors for $r^{2}$ are $r \cdot r$ , which is $r$ multiplied by each other $2$ times.

$r^{2} = r \cdot r$

$r$ occurs $2$ times.

Step 1.9

The LCM of $r^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$r \cdot r$

Step 1.10

Multiply $r$ by $r$ .

$r^{2}$

Step 1.11

The LCM for $r^{2}, 3, 1$ is the numeric part $3$ multiplied by the variable part.

$3 r^{2}$

Step 2

Multiply each term in $- \frac{12000}{r^{2}} + \frac{16 π r}{3} = 0$ by $3 r^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $- \frac{12000}{r^{2}} + \frac{16 π r}{3} = 0$ by $3 r^{2}$ .

$- \frac{12000}{r^{2}} (3 r^{2}) + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Move the leading negative in $- \frac{12000}{r^{2}}$ into the numerator.

$\frac{- 12000}{r^{2}} (3 r^{2}) + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2.1.1.2

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

$\frac{- 12000}{r^{2}} (r^{2} \cdot 3) + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2.1.1.3

Cancel the common factor.

$\frac{- 12000}{r^{2}} (r^{2} \cdot 3) + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2.1.1.4

Rewrite the expression.

$- 12000 \cdot 3 + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2.1.2

Multiply $- 12000$ by $3$ .

$- 36000 + \frac{16 π r}{3} (3 r^{2}) = 0 (3 r^{2})$

Step 2.2.1.3

Rewrite using the commutative property of multiplication.

$- 36000 + 3 \frac{16 π r}{3} r^{2} = 0 (3 r^{2})$

Step 2.2.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.4.1

Cancel the common factor.

$- 36000 + 3 \frac{16 π r}{3} r^{2} = 0 (3 r^{2})$

Step 2.2.1.4.2

Rewrite the expression.

$- 36000 + 16 π r \cdot r^{2} = 0 (3 r^{2})$

Step 2.2.1.5

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

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Step 2.2.1.5.1

Move $r^{2}$ .

$- 36000 + 16 π (r^{2} r) = 0 (3 r^{2})$

Step 2.2.1.5.2

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

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Step 2.2.1.5.2.1

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

$- 36000 + 16 π (r^{2} r^{1}) = 0 (3 r^{2})$

Step 2.2.1.5.2.2

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

$- 36000 + 16 π r^{2 + 1} = 0 (3 r^{2})$

Step 2.2.1.5.3

Add $2$ and $1$ .

$- 36000 + 16 π r^{3} = 0 (3 r^{2})$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0 (3 r^{2})$ .

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Step 2.3.1.1

Multiply $3$ by $0$ .

$- 36000 + 16 π r^{3} = 0 r^{2}$

Step 2.3.1.2

Multiply $0$ by $r^{2}$ .

$- 36000 + 16 π r^{3} = 0$

Step 3

Solve the equation.

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Step 3.1

Add $36000$ to both sides of the equation.

$16 π r^{3} = 36000$

Step 3.2

Subtract $36000$ from both sides of the equation.

$16 π r^{3} - 36000 = 0$

Step 3.3

Factor $16$ out of $16 π r^{3} - 36000$ .

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Step 3.3.1

Factor $16$ out of $16 π r^{3}$ .

$16 (π r^{3}) - 36000 = 0$

Step 3.3.2

Factor $16$ out of $- 36000$ .

$16 (π r^{3}) + 16 \cdot - 2250 = 0$

Step 3.3.3

Factor $16$ out of $16 (π r^{3}) + 16 \cdot - 2250$ .

$16 (π r^{3} - 2250) = 0$

Step 3.4

Divide each term in $16 (π r^{3} - 2250) = 0$ by $16$ and simplify.

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Step 3.4.1

Divide each term in $16 (π r^{3} - 2250) = 0$ by $16$ .

$\frac{16 (π r^{3} - 2250)}{16} = \frac{0}{16}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $16$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{16 (π r^{3} - 2250)}{16} = \frac{0}{16}$

Step 3.4.2.1.2

Divide $π r^{3} - 2250$ by $1$ .

$π r^{3} - 2250 = \frac{0}{16}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $16$ .

$π r^{3} - 2250 = 0$

Step 3.5

Add $2250$ to both sides of the equation.

$π r^{3} = 2250$

Step 3.6

Divide each term in $π r^{3} = 2250$ by $π$ and simplify.

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Step 3.6.1

Divide each term in $π r^{3} = 2250$ by $π$ .

$\frac{π r^{3}}{π} = \frac{2250}{π}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $π$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{π r^{3}}{π} = \frac{2250}{π}$

Step 3.6.2.1.2

Divide $r^{3}$ by $1$ .

$r^{3} = \frac{2250}{π}$

Step 3.7

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

$r = \sqrt[3]{\frac{2250}{π}}$

Step 3.8

Simplify $\sqrt[3]{\frac{2250}{π}}$ .

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Step 3.8.1

Rewrite $\sqrt[3]{\frac{2250}{π}}$ as $\frac{\sqrt[3]{2250}}{\sqrt[3]{π}}$ .

$r = \frac{\sqrt[3]{2250}}{\sqrt[3]{π}}$

Step 3.8.2

Simplify the numerator.

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Step 3.8.2.1

Rewrite $2250$ as $5^{3} \cdot 18$ .

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Step 3.8.2.1.1

Factor $125$ out of $2250$ .

$r = \frac{\sqrt[3]{125 (18)}}{\sqrt[3]{π}}$

Step 3.8.2.1.2

Rewrite $125$ as $5^{3}$ .

$r = \frac{\sqrt[3]{5^{3} \cdot 18}}{\sqrt[3]{π}}$

Step 3.8.2.2

Pull terms out from under the radical.

$r = \frac{5 \sqrt[3]{18}}{\sqrt[3]{π}}$

Step 3.8.3

Multiply $\frac{5 \sqrt[3]{18}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$r = \frac{5 \sqrt[3]{18}}{\sqrt[3]{π}} \cdot \frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$

Step 3.8.4

Combine and simplify the denominator.

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Step 3.8.4.1

Multiply $\frac{5 \sqrt[3]{18}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{\sqrt[3]{π} {\sqrt[3]{π}}^{2}}$

Step 3.8.4.2

Raise $\sqrt[3]{π}$ to the power of $1$ .

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1} {\sqrt[3]{π}}^{2}}$

Step 3.8.4.3

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

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1 + 2}}$

Step 3.8.4.4

Add $1$ and $2$ .

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{3}}$

Step 3.8.4.5

Rewrite ${\sqrt[3]{π}}^{3}$ as $π$ .

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Step 3.8.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{π}$ as $π^{\frac{1}{3}}$ .

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{{(π^{\frac{1}{3}})}^{3}}$

Step 3.8.4.5.2

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

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{π^{\frac{1}{3} \cdot 3}}$

Step 3.8.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Step 3.8.4.5.4

Cancel the common factor of $3$ .

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Step 3.8.4.5.4.1

Cancel the common factor.

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Step 3.8.4.5.4.2

Rewrite the expression.

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{π^{1}}$

Step 3.8.4.5.5

Simplify.

$r = \frac{5 \sqrt[3]{18} {\sqrt[3]{π}}^{2}}{π}$

Step 3.8.5

Simplify the numerator.

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Step 3.8.5.1

Rewrite ${\sqrt[3]{π}}^{2}$ as $\sqrt[3]{π^{2}}$ .

$r = \frac{5 \sqrt[3]{18} \sqrt[3]{π^{2}}}{π}$

Step 3.8.5.2

Combine using the product rule for radicals.

$r = \frac{5 \sqrt[3]{π^{2} \cdot 18}}{π}$

Step 3.8.6

Move $18$ to the left of $π^{2}$ .

$r = \frac{5 \sqrt[3]{18 π^{2}}}{π}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$r = \frac{5 \sqrt[3]{18 π^{2}}}{π}$

Decimal Form:

$r = 8.94700228 \dots$