Trigonometry Examples

$38000 = \frac{2 p r^{45.3}}{360}$

Step 1

Rewrite the equation as $\frac{2 p r^{45.3}}{360} = 38000$ .

$\frac{2 p r^{45.3}}{360} = 38000$

Step 2

Cancel the common factor of $2$ and $360$ .

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Step 2.1

Factor $2$ out of $2 p r^{45.3}$ .

$\frac{2 (p r^{45.3})}{360} = 38000$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $360$ .

$\frac{2 (p r^{45.3})}{2 \cdot 180} = 38000$

Step 2.2.2

Cancel the common factor.

$\frac{2 (p r^{45.3})}{2 \cdot 180} = 38000$

Step 2.2.3

Rewrite the expression.

$\frac{p r^{45.3}}{180} = 38000$

Step 3

Multiply both sides by $180$ .

$\frac{p r^{45.3}}{180} \cdot 180 = 38000 \cdot 180$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $180$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{p r^{45.3}}{180} \cdot 180 = 38000 \cdot 180$

Step 4.1.1.2

Rewrite the expression.

$p r^{45.3} = 38000 \cdot 180$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $38000$ by $180$ .

$p r^{45.3} = 6840000$

Step 5

Solve for $r$ .

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Step 5.1

Divide each term in $p r^{45.3} = 6840000$ by $p$ and simplify.

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Step 5.1.1

Divide each term in $p r^{45.3} = 6840000$ by $p$ .

$\frac{p r^{45.3}}{p} = \frac{6840000}{p}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $p$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{p r^{45.3}}{p} = \frac{6840000}{p}$

Step 5.1.2.1.2

Divide $r^{45.3}$ by $1$ .

$r^{45.3} = \frac{6840000}{p}$

Step 5.2

Convert the decimal exponent to a fractional exponent.

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Step 5.2.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$r^{45 \frac{3}{10}} = \frac{6840000}{p}$

Step 5.2.2

Convert $45 \frac{3}{10}$ to an improper fraction.

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Step 5.2.2.1

A mixed number is an addition of its whole and fractional parts.

$r^{45 + \frac{3}{10}} = \frac{6840000}{p}$

Step 5.2.2.2

Add $45$ and $\frac{3}{10}$ .

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Step 5.2.2.2.1

To write $45$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$r^{45 \cdot \frac{10}{10} + \frac{3}{10}} = \frac{6840000}{p}$

Step 5.2.2.2.2

Combine $45$ and $\frac{10}{10}$ .

$r^{\frac{45 \cdot 10}{10} + \frac{3}{10}} = \frac{6840000}{p}$

Step 5.2.2.2.3

Combine the numerators over the common denominator.

$r^{\frac{45 \cdot 10 + 3}{10}} = \frac{6840000}{p}$

Step 5.2.2.2.4

Simplify the numerator.

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Step 5.2.2.2.4.1

Multiply $45$ by $10$ .

$r^{\frac{450 + 3}{10}} = \frac{6840000}{p}$

Step 5.2.2.2.4.2

Add $450$ and $3$ .

$r^{\frac{453}{10}} = \frac{6840000}{p}$

Step 5.3

Raise each side of the equation to the power of $\frac{1}{45.3}$ to eliminate the fractional exponent on the left side.

${(r^{\frac{453}{10}})}^{\frac{1}{45.3}} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4

Simplify the exponent.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify ${(r^{\frac{453}{10}})}^{\frac{1}{45.3}}$ .

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Step 5.4.1.1.1

Multiply the exponents in ${(r^{\frac{453}{10}})}^{\frac{1}{45.3}}$ .

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Step 5.4.1.1.1.1

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

$r^{\frac{453}{10} \cdot \frac{1}{45.3}} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.1.1.1.2

Cancel the common factor of $45.3$ .

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Step 5.4.1.1.1.2.1

Factor $45.3$ out of $453$ .

$r^{\frac{45.3 (10)}{10} \cdot \frac{1}{45.3}} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.1.1.1.2.2

Cancel the common factor.

$r^{\frac{45.3 \cdot 10}{10} \cdot \frac{1}{45.3}} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.1.1.1.2.3

Rewrite the expression.

$r^{\frac{10}{10}} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.1.1.1.3

Divide $10$ by $10$ .

$r^{1} = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.1.1.2

Simplify.

$r = {(\frac{6840000}{p})}^{\frac{1}{45.3}}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify ${(\frac{6840000}{p})}^{\frac{1}{45.3}}$ .

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Step 5.4.2.1.1

Divide $1$ by $45.3$ .

$r = {(\frac{6840000}{p})}^{0.02207505}$

Step 5.4.2.1.2

Apply the product rule to $\frac{6840000}{p}$ .

$r = \frac{6840000^{0.02207505}}{p^{0.02207505}}$

Step 5.4.2.1.3

Raise $6840000$ to the power of $0.02207505$ .

$r = \frac{1.41541645}{p^{0.02207505}}$