Trigonometry Examples

$\frac{100}{1 + 25 e^{- 0.095 t}} = 70$

Step 1

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

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 70 (1 + 25 e^{- 0.095 t})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

$\frac{100}{1 + 25 e^{- 0.095 t}} (1 + 25 e^{- 0.095 t}) = 70 (1 + 25 e^{- 0.095 t})$

Step 2.1.1.2

Rewrite the expression.

$100 = 70 (1 + 25 e^{- 0.095 t})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $70 (1 + 25 e^{- 0.095 t})$ .

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Step 2.2.1.1

Apply the distributive property.

$100 = 70 \cdot 1 + 70 (25 e^{- 0.095 t})$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $70$ by $1$ .

$100 = 70 + 70 (25 e^{- 0.095 t})$

Step 2.2.1.2.2

Multiply $25$ by $70$ .

$100 = 70 + 1750 e^{- 0.095 t}$

Step 3

Solve for $t$ .

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Step 3.1

Rewrite the equation as $70 + 1750 e^{- 0.095 t} = 100$ .

$70 + 1750 e^{- 0.095 t} = 100$

Step 3.2

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

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Step 3.2.1

Subtract $70$ from both sides of the equation.

$1750 e^{- 0.095 t} = 100 - 70$

Step 3.2.2

Subtract $70$ from $100$ .

$1750 e^{- 0.095 t} = 30$

Step 3.3

Divide each term in $1750 e^{- 0.095 t} = 30$ by $1750$ and simplify.

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Step 3.3.1

Divide each term in $1750 e^{- 0.095 t} = 30$ by $1750$ .

$\frac{1750 e^{- 0.095 t}}{1750} = \frac{30}{1750}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $1750$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1750 e^{- 0.095 t}}{1750} = \frac{30}{1750}$

Step 3.3.2.1.2

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

$e^{- 0.095 t} = \frac{30}{1750}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $30$ and $1750$ .

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Step 3.3.3.1.1

Factor $10$ out of $30$ .

$e^{- 0.095 t} = \frac{10 (3)}{1750}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $10$ out of $1750$ .

$e^{- 0.095 t} = \frac{10 \cdot 3}{10 \cdot 175}$

Step 3.3.3.1.2.2

Cancel the common factor.

$e^{- 0.095 t} = \frac{10 \cdot 3}{10 \cdot 175}$

Step 3.3.3.1.2.3

Rewrite the expression.

$e^{- 0.095 t} = \frac{3}{175}$

Step 3.4

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

$\ln (e^{- 0.095 t}) = \ln (\frac{3}{175})$

Step 3.5

Expand the left side.

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Step 3.5.1

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

$- 0.095 t \ln (e) = \ln (\frac{3}{175})$

Step 3.5.2

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

$- 0.095 t \cdot 1 = \ln (\frac{3}{175})$

Step 3.5.3

Multiply $- 0.095$ by $1$ .

$- 0.095 t = \ln (\frac{3}{175})$

Step 3.6

Divide each term in $- 0.095 t = \ln (\frac{3}{175})$ by $- 0.095$ and simplify.

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Step 3.6.1

Divide each term in $- 0.095 t = \ln (\frac{3}{175})$ by $- 0.095$ .

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (\frac{3}{175})}{- 0.095}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $- 0.095$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{- 0.095 t}{- 0.095} = \frac{\ln (\frac{3}{175})}{- 0.095}$

Step 3.6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{3}{175})}{- 0.095}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{3}{175})}{0.095}$

Step 3.6.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{3}{175})}{0.095}$

Step 3.6.3.3

Divide $3$ by $175$ .

$t = - \frac{\log_{2.71828182} (0.01714285)}{0.095}$

Step 3.6.3.4

Log base $2.71828182$ of $0.01714285$ is approximately $- 4.06617368$ .

$t = - \frac{- 4.06617368}{0.095}$

Step 3.6.3.5

Divide $- 4.06617368$ by $0.095$ .

$t = - - 42.80182826$

Step 3.6.3.6

Multiply $- 1$ by $- 42.80182826$ .

$t = 42.80182826$