Trigonometry Examples

$80 = (89 - 75) e^{- 0.094 t} + 75$

Step 1

Rewrite the equation as $(89 - 75) e^{- 0.094 t} + 75 = 80$ .

$(89 - 75) e^{- 0.094 t} + 75 = 80$

Step 2

Subtract $75$ from $89$ .

$14 e^{- 0.094 t} + 75 = 80$

Step 3

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

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Step 3.1

Subtract $75$ from both sides of the equation.

$14 e^{- 0.094 t} = 80 - 75$

Step 3.2

Subtract $75$ from $80$ .

$14 e^{- 0.094 t} = 5$

Step 4

Divide each term in $14 e^{- 0.094 t} = 5$ by $14$ and simplify.

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Step 4.1

Divide each term in $14 e^{- 0.094 t} = 5$ by $14$ .

$\frac{14 e^{- 0.094 t}}{14} = \frac{5}{14}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $14$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{14 e^{- 0.094 t}}{14} = \frac{5}{14}$

Step 4.2.1.2

Divide $e^{- 0.094 t}$ by $1$ .

$e^{- 0.094 t} = \frac{5}{14}$

Step 5

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

$\ln (e^{- 0.094 t}) = \ln (\frac{5}{14})$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.094 t})$ by moving $- 0.094 t$ outside the logarithm.

$- 0.094 t \ln (e) = \ln (\frac{5}{14})$

Step 6.2

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

$- 0.094 t \cdot 1 = \ln (\frac{5}{14})$

Step 6.3

Multiply $- 0.094$ by $1$ .

$- 0.094 t = \ln (\frac{5}{14})$

Step 7

Divide each term in $- 0.094 t = \ln (\frac{5}{14})$ by $- 0.094$ and simplify.

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Step 7.1

Divide each term in $- 0.094 t = \ln (\frac{5}{14})$ by $- 0.094$ .

$\frac{- 0.094 t}{- 0.094} = \frac{\ln (\frac{5}{14})}{- 0.094}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.094$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.094 t}{- 0.094} = \frac{\ln (\frac{5}{14})}{- 0.094}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{5}{14})}{- 0.094}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{5}{14})}{0.094}$

Step 7.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{5}{14})}{0.094}$

Step 7.3.3

Divide $5$ by $14$ .

$t = - \frac{\log_{2.71828182} (0.3\overline{571428})}{0.094}$

Step 7.3.4

Log base $2.71828182$ of $0.3\overline{571428}$ is approximately $- 1.02961941$ .

$t = - \frac{- 1.02961941}{0.094}$

Step 7.3.5

Divide $- 1.02961941$ by $0.094$ .

$t = - - 10.95339805$

Step 7.3.6

Multiply $- 1$ by $- 10.95339805$ .

$t = 10.95339805$