Finite Math Examples

$\frac{50}{1 + 4 e^{0.2 t}} = 20$

Step 1

Multiply both sides by $1 + 4 e^{0.2 t}$ .

$\frac{50}{1 + 4 e^{0.2 t}} (1 + 4 e^{0.2 t}) = 20 (1 + 4 e^{0.2 t})$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $1 + 4 e^{0.2 t}$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$\frac{50}{1 + 4 e^{0.2 t}} (1 + 4 e^{0.2 t}) = 20 (1 + 4 e^{0.2 t})$

Step 2.1.1.2

Rewrite the expression.

$50 = 20 (1 + 4 e^{0.2 t})$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $20 (1 + 4 e^{0.2 t})$ .

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$50 = 20 \cdot 1 + 20 (4 e^{0.2 t})$

Step 2.2.1.2

Multiply.

Tap for more steps...

Step 2.2.1.2.1

Multiply $20$ by $1$ .

$50 = 20 + 20 (4 e^{0.2 t})$

Step 2.2.1.2.2

Multiply $4$ by $20$ .

$50 = 20 + 80 e^{0.2 t}$

Step 3

Solve for $t$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $20 + 80 e^{0.2 t} = 50$ .

$20 + 80 e^{0.2 t} = 50$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $20$ from both sides of the equation.

$80 e^{0.2 t} = 50 - 20$

Step 3.2.2

Subtract $20$ from $50$ .

$80 e^{0.2 t} = 30$

Step 3.3

Divide each term in $80 e^{0.2 t} = 30$ by $80$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $80 e^{0.2 t} = 30$ by $80$ .

$\frac{80 e^{0.2 t}}{80} = \frac{30}{80}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $80$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{80 e^{0.2 t}}{80} = \frac{30}{80}$

Step 3.3.2.1.2

Divide $e^{0.2 t}$ by $1$ .

$e^{0.2 t} = \frac{30}{80}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Cancel the common factor of $30$ and $80$ .

Tap for more steps...

Step 3.3.3.1.1

Factor $10$ out of $30$ .

$e^{0.2 t} = \frac{10 (3)}{80}$

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $10$ out of $80$ .

$e^{0.2 t} = \frac{10 \cdot 3}{10 \cdot 8}$

Step 3.3.3.1.2.2

Cancel the common factor.

$e^{0.2 t} = \frac{10 \cdot 3}{10 \cdot 8}$

Step 3.3.3.1.2.3

Rewrite the expression.

$e^{0.2 t} = \frac{3}{8}$

Step 3.4

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

$\ln (e^{0.2 t}) = \ln (\frac{3}{8})$

Step 3.5

Expand the left side.

Tap for more steps...

Step 3.5.1

Expand $\ln (e^{0.2 t})$ by moving $0.2 t$ outside the logarithm.

$0.2 t \ln (e) = \ln (\frac{3}{8})$

Step 3.5.2

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

$0.2 t \cdot 1 = \ln (\frac{3}{8})$

Step 3.5.3

Multiply $0.2$ by $1$ .

$0.2 t = \ln (\frac{3}{8})$

Step 3.6

Divide each term in $0.2 t = \ln (\frac{3}{8})$ by $0.2$ and simplify.

Tap for more steps...

Step 3.6.1

Divide each term in $0.2 t = \ln (\frac{3}{8})$ by $0.2$ .

$\frac{0.2 t}{0.2} = \frac{\ln (\frac{3}{8})}{0.2}$

Step 3.6.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.1

Cancel the common factor of $0.2$ .

Tap for more steps...

Step 3.6.2.1.1

Cancel the common factor.

$\frac{0.2 t}{0.2} = \frac{\ln (\frac{3}{8})}{0.2}$

Step 3.6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{3}{8})}{0.2}$

Step 3.6.3

Simplify the right side.

Tap for more steps...

Step 3.6.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{3}{8})}{0.2}$

Step 3.6.3.2

Divide $3$ by $8$ .

$t = \frac{\log_{2.71828182} (0.375)}{0.2}$

Step 3.6.3.3

Log base $2.71828182$ of $0.375$ is approximately $- 0.98082925$ .

$t = \frac{- 0.98082925}{0.2}$

Step 3.6.3.4

Divide $- 0.98082925$ by $0.2$ .

$t = - 4.90414626$