Calculus Examples

$c (t) = 3 t e^{- \frac{1}{30} t}$

Step 1

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

$3 t e^{- \frac{1}{30} t} = c t$

Step 2

Simplify $3 t e^{- \frac{1}{30} t}$ .

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Step 2.1

Rewrite.

$0 + 0 + 3 t e^{- \frac{1}{30} t} = c t$

Step 2.2

Simplify by adding zeros.

$3 t e^{- \frac{1}{30} t} = c t$

Step 2.3

Combine $t$ and $\frac{1}{30}$ .

$3 t e^{- \frac{t}{30}} = c t$

Step 3

Subtract $c t$ from both sides of the equation.

$3 t e^{- \frac{t}{30}} - c t = 0$

Step 4

Factor $t$ out of $3 t e^{- \frac{t}{30}} - c t$ .

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Step 4.1

Factor $t$ out of $3 t e^{- \frac{t}{30}}$ .

$t (3 e^{- \frac{t}{30}}) - c t = 0$

Step 4.2

Factor $t$ out of $- c t$ .

$t (3 e^{- \frac{t}{30}}) + t (- c) = 0$

Step 4.3

Factor $t$ out of $t (3 e^{- \frac{t}{30}}) + t (- c)$ .

$t (3 e^{- \frac{t}{30}} - c) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t = 0$

$3 e^{- \frac{t}{30}} - c = 0$

Step 6

Set $t$ equal to $0$ .

$t = 0$

Step 7

Set $3 e^{- \frac{t}{30}} - c$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $3 e^{- \frac{t}{30}} - c$ equal to $0$ .

$3 e^{- \frac{t}{30}} - c = 0$

Step 7.2

Solve $3 e^{- \frac{t}{30}} - c = 0$ for $t$ .

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Step 7.2.1

Add $c$ to both sides of the equation.

$3 e^{- \frac{t}{30}} = c$

Step 7.2.2

Divide each term in $3 e^{- \frac{t}{30}} = c$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 e^{- \frac{t}{30}} = c$ by $3$ .

$\frac{3 e^{- \frac{t}{30}}}{3} = \frac{c}{3}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 e^{- \frac{t}{30}}}{3} = \frac{c}{3}$

Step 7.2.2.2.1.2

Divide $e^{- \frac{t}{30}}$ by $1$ .

$e^{- \frac{t}{30}} = \frac{c}{3}$

Step 7.2.3

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

$\ln (e^{- \frac{t}{30}}) = \ln (\frac{c}{3})$

Step 7.2.4

Expand the left side.

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Step 7.2.4.1

Expand $\ln (e^{- \frac{t}{30}})$ by moving $- \frac{t}{30}$ outside the logarithm.

$- \frac{t}{30} \ln (e) = \ln (\frac{c}{3})$

Step 7.2.4.2

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

$- \frac{t}{30} \cdot 1 = \ln (\frac{c}{3})$

Step 7.2.4.3

Multiply $- 1$ by $1$ .

$- \frac{t}{30} = \ln (\frac{c}{3})$

Step 7.2.5

Multiply both sides of the equation by $- 30$ .

$- 30 (- \frac{t}{30}) = - 30 \ln (\frac{c}{3})$

Step 7.2.6

Simplify the left side.

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Step 7.2.6.1

Simplify $- 30 (- \frac{t}{30})$ .

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Step 7.2.6.1.1

Cancel the common factor of $30$ .

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Step 7.2.6.1.1.1

Move the leading negative in $- \frac{t}{30}$ into the numerator.

$- 30 \frac{- t}{30} = - 30 \ln (\frac{c}{3})$

Step 7.2.6.1.1.2

Factor $30$ out of $- 30$ .

$30 (- 1) \frac{- t}{30} = - 30 \ln (\frac{c}{3})$

Step 7.2.6.1.1.3

Cancel the common factor.

$30 \cdot - 1 \frac{- t}{30} = - 30 \ln (\frac{c}{3})$

Step 7.2.6.1.1.4

Rewrite the expression.

$- - t = - 30 \ln (\frac{c}{3})$

Step 7.2.6.1.2

Multiply.

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Step 7.2.6.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 30 \ln (\frac{c}{3})$

Step 7.2.6.1.2.2

Multiply $t$ by $1$ .

$t = - 30 \ln (\frac{c}{3})$

Step 8

The final solution is all the values that make $t (3 e^{- \frac{t}{30}} - c) = 0$ true.

$t = 0$

$t = - 30 \ln (\frac{c}{3})$