Trigonometry Examples

$140 = (165 - 70) e^{- 0.046 (t)} + 70$

Step 1

Rewrite the equation as $(165 - 70) e^{- 0.046 t} + 70 = 140$ .

$(165 - 70) e^{- 0.046 t} + 70 = 140$

Step 2

Subtract $70$ from $165$ .

$95 e^{- 0.046 t} + 70 = 140$

Step 3

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

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Step 3.1

Subtract $70$ from both sides of the equation.

$95 e^{- 0.046 t} = 140 - 70$

Step 3.2

Subtract $70$ from $140$ .

$95 e^{- 0.046 t} = 70$

Step 4

Divide each term in $95 e^{- 0.046 t} = 70$ by $95$ and simplify.

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Step 4.1

Divide each term in $95 e^{- 0.046 t} = 70$ by $95$ .

$\frac{95 e^{- 0.046 t}}{95} = \frac{70}{95}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $95$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{95 e^{- 0.046 t}}{95} = \frac{70}{95}$

Step 4.2.1.2

Divide $e^{- 0.046 t}$ by $1$ .

$e^{- 0.046 t} = \frac{70}{95}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $70$ and $95$ .

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Step 4.3.1.1

Factor $5$ out of $70$ .

$e^{- 0.046 t} = \frac{5 (14)}{95}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $5$ out of $95$ .

$e^{- 0.046 t} = \frac{5 \cdot 14}{5 \cdot 19}$

Step 4.3.1.2.2

Cancel the common factor.

$e^{- 0.046 t} = \frac{5 \cdot 14}{5 \cdot 19}$

Step 4.3.1.2.3

Rewrite the expression.

$e^{- 0.046 t} = \frac{14}{19}$

Step 5

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

$\ln (e^{- 0.046 t}) = \ln (\frac{14}{19})$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.046 t})$ by moving $- 0.046 t$ outside the logarithm.

$- 0.046 t \ln (e) = \ln (\frac{14}{19})$

Step 6.2

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

$- 0.046 t \cdot 1 = \ln (\frac{14}{19})$

Step 6.3

Multiply $- 0.046$ by $1$ .

$- 0.046 t = \ln (\frac{14}{19})$

Step 7

Divide each term in $- 0.046 t = \ln (\frac{14}{19})$ by $- 0.046$ and simplify.

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Step 7.1

Divide each term in $- 0.046 t = \ln (\frac{14}{19})$ by $- 0.046$ .

$\frac{- 0.046 t}{- 0.046} = \frac{\ln (\frac{14}{19})}{- 0.046}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.046$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.046 t}{- 0.046} = \frac{\ln (\frac{14}{19})}{- 0.046}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{14}{19})}{- 0.046}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{14}{19})}{0.046}$

Step 7.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{14}{19})}{0.046}$

Step 7.3.3

Divide $14$ by $19$ .

$t = - \frac{\log_{2.71828182} (0.7368421)}{0.046}$

Step 7.3.4

Log base $2.71828182$ of $0.7368421$ is approximately $- 0.30538164$ .

$t = - \frac{- 0.30538164}{0.046}$

Step 7.3.5

Divide $- 0.30538164$ by $0.046$ .

$t = - - 6.63873151$

Step 7.3.6

Multiply $- 1$ by $- 6.63873151$ .

$t = 6.63873151$