Trigonometry Examples

$60 = \frac{100}{1 + 25 e^{- 0.095 t}}$

Step 1

Rewrite the equation as $\frac{100}{1 + 25 e^{- 0.095 t}} = 60$ .

$\frac{100}{1 + 25 e^{- 0.095 t}} = 60$

Step 2

Multiply both sides by $1 + 25 e^{- 0.095 t}$ .

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 60 (1 + 25 e^{- 0.095 t})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $1 + 25 e^{- 0.095 t}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 60 (1 + 25 e^{- 0.095 t})$

Step 3.1.1.2

Rewrite the expression.

$100 = 60 (1 + 25 e^{- 0.095 t})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $60 (1 + 25 e^{- 0.095 t})$ .

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Step 3.2.1.1

Apply the distributive property.

$100 = 60 \cdot 1 + 60 (25 e^{- 0.095 t})$

Step 3.2.1.2

Multiply.

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Step 3.2.1.2.1

Multiply $60$ by $1$ .

$100 = 60 + 60 (25 e^{- 0.095 t})$

Step 3.2.1.2.2

Multiply $25$ by $60$ .

$100 = 60 + 1500 e^{- 0.095 t}$

Step 4

Solve for $t$ .

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Step 4.1

Rewrite the equation as $60 + 1500 e^{- 0.095 t} = 100$ .

$60 + 1500 e^{- 0.095 t} = 100$

Step 4.2

Move all terms not containing $t$ to the right side of the equation.

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Step 4.2.1

Subtract $60$ from both sides of the equation.

$1500 e^{- 0.095 t} = 100 - 60$

Step 4.2.2

Subtract $60$ from $100$ .

$1500 e^{- 0.095 t} = 40$

Step 4.3

Divide each term in $1500 e^{- 0.095 t} = 40$ by $1500$ and simplify.

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Step 4.3.1

Divide each term in $1500 e^{- 0.095 t} = 40$ by $1500$ .

$\frac{1500 e^{- 0.095 t}}{1500} = \frac{40}{1500}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $1500$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{1500 e^{- 0.095 t}}{1500} = \frac{40}{1500}$

Step 4.3.2.1.2

Divide $e^{- 0.095 t}$ by $1$ .

$e^{- 0.095 t} = \frac{40}{1500}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $40$ and $1500$ .

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Step 4.3.3.1.1

Factor $20$ out of $40$ .

$e^{- 0.095 t} = \frac{20 (2)}{1500}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $20$ out of $1500$ .

$e^{- 0.095 t} = \frac{20 \cdot 2}{20 \cdot 75}$

Step 4.3.3.1.2.2

Cancel the common factor.

$e^{- 0.095 t} = \frac{20 \cdot 2}{20 \cdot 75}$

Step 4.3.3.1.2.3

Rewrite the expression.

$e^{- 0.095 t} = \frac{2}{75}$

Step 4.4

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

$\ln (e^{- 0.095 t}) = \ln (\frac{2}{75})$

Step 4.5

Expand the left side.

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Step 4.5.1

Expand $\ln (e^{- 0.095 t})$ by moving $- 0.095 t$ outside the logarithm.

$- 0.095 t \ln (e) = \ln (\frac{2}{75})$

Step 4.5.2

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

$- 0.095 t \cdot 1 = \ln (\frac{2}{75})$

Step 4.5.3

Multiply $- 0.095$ by $1$ .

$- 0.095 t = \ln (\frac{2}{75})$

Step 4.6

Divide each term in $- 0.095 t = \ln (\frac{2}{75})$ by $- 0.095$ and simplify.

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Step 4.6.1

Divide each term in $- 0.095 t = \ln (\frac{2}{75})$ by $- 0.095$ .

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (\frac{2}{75})}{- 0.095}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $- 0.095$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (\frac{2}{75})}{- 0.095}$

Step 4.6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{2}{75})}{- 0.095}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{2}{75})}{0.095}$

Step 4.6.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{2}{75})}{0.095}$

Step 4.6.3.3

Divide $2$ by $75$ .

$t = - \frac{\log_{2.71828182} (0.02\overline{6})}{0.095}$

Step 4.6.3.4

Log base $2.71828182$ of $0.02\overline{6}$ is approximately $- 3.62434093$ .

$t = - \frac{- 3.62434093}{0.095}$

Step 4.6.3.5

Divide $- 3.62434093$ by $0.095$ .

$t = - - 38.15095718$

Step 4.6.3.6

Multiply $- 1$ by $- 38.15095718$ .

$t = 38.15095718$