Trigonometry Examples

$0.85 = \frac{100}{1 + 25 e^{- 0.095 t}}$

Step 1

Rewrite the equation as $\frac{100}{1 + 25 e^{- 0.095 t}} = 0.85$ .

$\frac{100}{1 + 25 e^{- 0.095 t}} = 0.85$

Step 2

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

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 0.85 (1 + 25 e^{- 0.095 t})$

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.

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 0.85 (1 + 25 e^{- 0.095 t})$

Step 3.1.1.2

Rewrite the expression.

$100 = 0.85 (1 + 25 e^{- 0.095 t})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $0.85 (1 + 25 e^{- 0.095 t})$ .

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Step 3.2.1.1

Apply the distributive property.

$100 = 0.85 \cdot 1 + 0.85 (25 e^{- 0.095 t})$

Step 3.2.1.2

Multiply.

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Step 3.2.1.2.1

Multiply $0.85$ by $1$ .

$100 = 0.85 + 0.85 (25 e^{- 0.095 t})$

Step 3.2.1.2.2

Multiply $25$ by $0.85$ .

$100 = 0.85 + 21.25 e^{- 0.095 t}$

Step 4

Solve for $t$ .

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Step 4.1

Rewrite the equation as $0.85 + 21.25 e^{- 0.095 t} = 100$ .

$0.85 + 21.25 e^{- 0.095 t} = 100$

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 $0.85$ from both sides of the equation.

$21.25 e^{- 0.095 t} = 100 - 0.85$

Step 4.2.2

Subtract $0.85$ from $100$ .

$21.25 e^{- 0.095 t} = 99.15$

Step 4.3

Divide each term in $21.25 e^{- 0.095 t} = 99.15$ by $21.25$ and simplify.

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Step 4.3.1

Divide each term in $21.25 e^{- 0.095 t} = 99.15$ by $21.25$ .

$\frac{21.25 e^{- 0.095 t}}{21.25} = \frac{99.15}{21.25}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $21.25$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{21.25 e^{- 0.095 t}}{21.25} = \frac{99.15}{21.25}$

Step 4.3.2.1.2

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

$e^{- 0.095 t} = \frac{99.15}{21.25}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $99.15$ by $21.25$ .

$e^{- 0.095 t} = 4.66588235$

Step 4.4

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

$\ln (e^{- 0.095 t}) = \ln (4.66588235)$

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.

$- 0.095 t \ln (e) = \ln (4.66588235)$

Step 4.5.2

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

$- 0.095 t \cdot 1 = \ln (4.66588235)$

Step 4.5.3

Multiply $- 0.095$ by $1$ .

$- 0.095 t = \ln (4.66588235)$

Step 4.6

Divide each term in $- 0.095 t = \ln (4.66588235)$ by $- 0.095$ and simplify.

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Step 4.6.1

Divide each term in $- 0.095 t = \ln (4.66588235)$ by $- 0.095$ .

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (4.66588235)}{- 0.095}$

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.

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (4.66588235)}{- 0.095}$

Step 4.6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (4.66588235)}{- 0.095}$

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.

$t = - \frac{\ln (4.66588235)}{0.095}$

Step 4.6.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (4.66588235)}{0.095}$

Step 4.6.3.3

Log base $2.71828182$ of $4.66588235$ is approximately $1.54027695$ .

$t = - \frac{1.54027695}{0.095}$

Step 4.6.3.4

Divide $1.54027695$ by $0.095$ .

$t = - 1 \cdot 16.21344167$

Step 4.6.3.5

Multiply $- 1$ by $16.21344167$ .

$t = - 16.21344167$