Trigonometry Examples

$1800 = 2600 {(1 - 0.51 e^{- 0.075 t})}^{3}$

Step 1

Rewrite the equation as $2600 {(1 - 0.51 e^{- 0.075 t})}^{3} = 1800$ .

$2600 {(1 - 0.51 e^{- 0.075 t})}^{3} = 1800$

Step 2

Divide each term in $2600 {(1 - 0.51 e^{- 0.075 t})}^{3} = 1800$ by $2600$ and simplify.

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Step 2.1

Divide each term in $2600 {(1 - 0.51 e^{- 0.075 t})}^{3} = 1800$ by $2600$ .

$\frac{2600 {(1 - 0.51 e^{- 0.075 t})}^{3}}{2600} = \frac{1800}{2600}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2600$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2600 {(1 - 0.51 e^{- 0.075 t})}^{3}}{2600} = \frac{1800}{2600}$

Step 2.2.1.2

Divide ${(1 - 0.51 e^{- 0.075 t})}^{3}$ by $1$ .

${(1 - 0.51 e^{- 0.075 t})}^{3} = \frac{1800}{2600}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $1800$ and $2600$ .

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Step 2.3.1.1

Factor $200$ out of $1800$ .

${(1 - 0.51 e^{- 0.075 t})}^{3} = \frac{200 (9)}{2600}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $200$ out of $2600$ .

${(1 - 0.51 e^{- 0.075 t})}^{3} = \frac{200 \cdot 9}{200 \cdot 13}$

Step 2.3.1.2.2

Cancel the common factor.

${(1 - 0.51 e^{- 0.075 t})}^{3} = \frac{200 \cdot 9}{200 \cdot 13}$

Step 2.3.1.2.3

Rewrite the expression.

${(1 - 0.51 e^{- 0.075 t})}^{3} = \frac{9}{13}$

Step 3

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

$1 - 0.51 e^{- 0.075 t} = \sqrt[3]{\frac{9}{13}}$

Step 4

Simplify $\sqrt[3]{\frac{9}{13}}$ .

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Step 4.1

Rewrite $\sqrt[3]{\frac{9}{13}}$ as $\frac{\sqrt[3]{9}}{\sqrt[3]{13}}$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9}}{\sqrt[3]{13}}$

Step 4.2

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{13}}$ by $\frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9}}{\sqrt[3]{13}} \cdot \frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{13}}$ by $\frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{\sqrt[3]{13} {\sqrt[3]{13}}^{2}}$

Step 4.3.2

Raise $\sqrt[3]{13}$ to the power of $1$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{1} {\sqrt[3]{13}}^{2}}$

Step 4.3.3

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

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{1 + 2}}$

Step 4.3.4

Add $1$ and $2$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{3}}$

Step 4.3.5

Rewrite ${\sqrt[3]{13}}^{3}$ as $13$ .

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Step 4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{13}$ as $13^{\frac{1}{3}}$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{{(13^{\frac{1}{3}})}^{3}}$

Step 4.3.5.2

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

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{13^{\frac{1}{3} \cdot 3}}$

Step 4.3.5.3

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

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{13^{\frac{3}{3}}}$

Step 4.3.5.4

Cancel the common factor of $3$ .

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Step 4.3.5.4.1

Cancel the common factor.

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{13^{\frac{3}{3}}}$

Step 4.3.5.4.2

Rewrite the expression.

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{13^{1}}$

Step 4.3.5.5

Evaluate the exponent.

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} {\sqrt[3]{13}}^{2}}{13}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Rewrite ${\sqrt[3]{13}}^{2}$ as $\sqrt[3]{13^{2}}$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} \sqrt[3]{13^{2}}}{13}$

Step 4.4.2

Raise $13$ to the power of $2$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9} \sqrt[3]{169}}{13}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Combine using the product rule for radicals.

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{9 \cdot 169}}{13}$

Step 4.5.2

Multiply $9$ by $169$ .

$1 - 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13}$

Step 5

Subtract $1$ from both sides of the equation.

$- 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13} - 1$

Step 6

Divide each term in $- 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13} - 1$ by $- 0.51$ and simplify.

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Step 6.1

Divide each term in $- 0.51 e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13} - 1$ by $- 0.51$ .

$\frac{- 0.51 e^{- 0.075 t}}{- 0.51} = \frac{\frac{\sqrt[3]{1521}}{13}}{- 0.51} + \frac{- 1}{- 0.51}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 0.51$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 0.51 e^{- 0.075 t}}{- 0.51} = \frac{\frac{\sqrt[3]{1521}}{13}}{- 0.51} + \frac{- 1}{- 0.51}$

Step 6.2.1.2

Divide $e^{- 0.075 t}$ by $1$ .

$e^{- 0.075 t} = \frac{\frac{\sqrt[3]{1521}}{13}}{- 0.51} + \frac{- 1}{- 0.51}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13} \cdot \frac{1}{- 0.51} + \frac{- 1}{- 0.51}$

Step 6.3.1.2

Divide $1$ by $- 0.51$ .

$e^{- 0.075 t} = \frac{\sqrt[3]{1521}}{13} \cdot - 1.96078431 + \frac{- 1}{- 0.51}$

Step 6.3.1.3

Multiply $\frac{\sqrt[3]{1521}}{13} \cdot - 1.96078431$ .

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Step 6.3.1.3.1

Combine $\frac{\sqrt[3]{1521}}{13}$ and $- 1.96078431$ .

$e^{- 0.075 t} = \frac{\sqrt[3]{1521} \cdot - 1.96078431}{13} + \frac{- 1}{- 0.51}$

Step 6.3.1.3.2

Multiply $\sqrt[3]{1521}$ by $- 1.96078431$ .

$e^{- 0.075 t} = \frac{- 22.54963735}{13} + \frac{- 1}{- 0.51}$

Step 6.3.1.4

Divide $- 22.54963735$ by $13$ .

$e^{- 0.075 t} = - 1.73458748 + \frac{- 1}{- 0.51}$

Step 6.3.1.5

Divide $- 1$ by $- 0.51$ .

$e^{- 0.075 t} = - 1.73458748 + 1.96078431$

Step 6.3.2

Add $- 1.73458748$ and $1.96078431$ .

$e^{- 0.075 t} = 0.22619682$

Step 7

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 0.075 t}) = \ln (0.22619682)$

Step 8

Expand the left side.

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Step 8.1

Expand $\ln (e^{- 0.075 t})$ by moving $- 0.075 t$ outside the logarithm.

$- 0.075 t \ln (e) = \ln (0.22619682)$

Step 8.2

The natural logarithm of $e$ is $1$ .

$- 0.075 t \cdot 1 = \ln (0.22619682)$

Step 8.3

Multiply $- 0.075$ by $1$ .

$- 0.075 t = \ln (0.22619682)$

Step 9

Divide each term in $- 0.075 t = \ln (0.22619682)$ by $- 0.075$ and simplify.

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Step 9.1

Divide each term in $- 0.075 t = \ln (0.22619682)$ by $- 0.075$ .

$\frac{- 0.075 t}{- 0.075} = \frac{\ln (0.22619682)}{- 0.075}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 0.075$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 0.075 t}{- 0.075} = \frac{\ln (0.22619682)}{- 0.075}$

Step 9.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.22619682)}{- 0.075}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.22619682)}{0.075}$

Step 9.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.22619682)}{0.075}$

Step 9.3.3

Log base $2.71828182$ of $0.22619682$ is approximately $- 1.48634975$ .

$t = - \frac{- 1.48634975}{0.075}$

Step 9.3.4

Divide $- 1.48634975$ by $0.075$ .

$t = - - 19.81799669$

Step 9.3.5

Multiply $- 1$ by $- 19.81799669$ .

$t = 19.81799669$