Calculus Examples

$\frac{400000}{50 + 7950 e^{- 0.5 t}} t = 12.9$

Step 1

Since $e$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$12.9 - \frac{8000 t}{1 + 159 e^{- 0.5 t}} = 0$

Step 2

Simplify $- 12.9 + \frac{8000 t}{1 + 159 e^{- 0.5 t}}$ .

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Step 2.1

To write $- 12.9$ as a fraction with a common denominator, multiply by $\frac{1 + 159 e^{- 0.5 t}}{1 + 159 e^{- 0.5 t}}$ .

$0 = - 12.9 \cdot \frac{1 + 159 e^{- 0.5 t}}{1 + 159 e^{- 0.5 t}} + \frac{8000 t}{1 + 159 e^{- 0.5 t}}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $- 12.9$ and $\frac{1 + 159 e^{- 0.5 t}}{1 + 159 e^{- 0.5 t}}$ .

$0 = \frac{- 12.9 (1 + 159 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}} + \frac{8000 t}{1 + 159 e^{- 0.5 t}}$

Step 2.2.2

Combine the numerators over the common denominator.

$0 = \frac{- 12.9 (1 + 159 e^{- 0.5 t}) + 8000 t}{1 + 159 e^{- 0.5 t}}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Factor $0.1$ out of $- 12.9 (1 + 159 e^{- 0.5 t}) + 8000 t$ .

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Step 2.3.1.1

Factor $0.1$ out of $- 12.9 (1 + 159 e^{- 0.5 t})$ .

$0 = \frac{0.1 (- 129 (1 + 159 e^{- 0.5 t})) + 8000 t}{1 + 159 e^{- 0.5 t}}$

Step 2.3.1.2

Factor $0.1$ out of $8000 t$ .

$0 = \frac{0.1 (- 129 (1 + 159 e^{- 0.5 t})) + 0.1 (80000 t)}{1 + 159 e^{- 0.5 t}}$

Step 2.3.1.3

Factor $0.1$ out of $0.1 (- 129 (1 + 159 e^{- 0.5 t})) + 0.1 (80000 t)$ .

$0 = \frac{0.1 (- 129 (1 + 159 e^{- 0.5 t}) + 80000 t)}{1 + 159 e^{- 0.5 t}}$

Step 2.3.2

Apply the distributive property.

$0 = \frac{0.1 (- 129 \cdot 1 - 129 (159 e^{- 0.5 t}) + 80000 t)}{1 + 159 e^{- 0.5 t}}$

Step 2.3.3

Multiply $- 129$ by $1$ .

$0 = \frac{0.1 (- 129 - 129 (159 e^{- 0.5 t}) + 80000 t)}{1 + 159 e^{- 0.5 t}}$

Step 2.3.4

Multiply $159$ by $- 129$ .

$0 = \frac{0.1 (- 129 - 20511 e^{- 0.5 t} + 80000 t)}{1 + 159 e^{- 0.5 t}}$

Step 2.3.5

Reorder terms.

$0 = \frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}}$

Step 3

Since $e$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}} = 0$

Step 4

Subtract $\frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}}$ from both sides of the equation.

$0 = - \frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}}$

Step 5

Rewrite the equation as $- \frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}} = 0$ .

$- \frac{0.1 (80000 t - 129 - 20511 e^{- 0.5 t})}{1 + 159 e^{- 0.5 t}} = 0$