Trigonometry Examples

$0.45 = 2800000 (1 - e^{- 0.03 t})$

Step 1

Rewrite the equation as $2800000 (1 - e^{- 0.03 t}) = 0.45$ .

$2800000 (1 - e^{- 0.03 t}) = 0.45$

Step 2

Divide each term in $2800000 (1 - e^{- 0.03 t}) = 0.45$ by $2800000$ and simplify.

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Step 2.1

Divide each term in $2800000 (1 - e^{- 0.03 t}) = 0.45$ by $2800000$ .

$\frac{2800000 (1 - e^{- 0.03 t})}{2800000} = \frac{0.45}{2800000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2800000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2800000 (1 - e^{- 0.03 t})}{2800000} = \frac{0.45}{2800000}$

Step 2.2.1.2

Divide $1 - e^{- 0.03 t}$ by $1$ .

$1 - e^{- 0.03 t} = \frac{0.45}{2800000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0.45$ by $2800000$ .

$1 - e^{- 0.03 t} = 0.00000016$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$- e^{- 0.03 t} = 0.00000016 - 1$

Step 3.2

Subtract $1$ from $0.00000016$ .

$- e^{- 0.03 t} = - 0.99999983$

Step 4

Divide each term in $- e^{- 0.03 t} = - 0.99999983$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{- 0.03 t} = - 0.99999983$ by $- 1$ .

$\frac{- e^{- 0.03 t}}{- 1} = \frac{- 0.99999983}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.03 t}}{1} = \frac{- 0.99999983}{- 1}$

Step 4.2.2

Divide $e^{- 0.03 t}$ by $1$ .

$e^{- 0.03 t} = \frac{- 0.99999983}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 0.99999983$ by $- 1$ .

$e^{- 0.03 t} = 0.99999983$

Step 5

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

$\ln (e^{- 0.03 t}) = \ln (0.99999983)$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.03 t})$ by moving $- 0.03 t$ outside the logarithm.

$- 0.03 t \ln (e) = \ln (0.99999983)$

Step 6.2

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

$- 0.03 t \cdot 1 = \ln (0.99999983)$

Step 6.3

Multiply $- 0.03$ by $1$ .

$- 0.03 t = \ln (0.99999983)$

Step 7

Divide each term in $- 0.03 t = \ln (0.99999983)$ by $- 0.03$ and simplify.

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Step 7.1

Divide each term in $- 0.03 t = \ln (0.99999983)$ by $- 0.03$ .

$\frac{- 0.03 t}{- 0.03} = \frac{\ln (0.99999983)}{- 0.03}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.03$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.03 t}{- 0.03} = \frac{\ln (0.99999983)}{- 0.03}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.99999983)}{- 0.03}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.99999983)}{0.03}$

Step 7.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.99999983)}{0.03}$

Step 7.3.3

Log base $2.71828182$ of $0.99999983$ is approximately $- 0.00000016$ .

$t = - \frac{- 0.00000016}{0.03}$

Step 7.3.4

Divide $- 0.00000016$ by $0.03$ .

$t = - - 0.00000535$

Step 7.3.5

Multiply $- 1$ by $- 0.00000535$ .

$t = 0.00000535$