Finite Math Examples

$C (t) = 60 e^{- 0.14 t}$

Step 1

Write $C (t) = 60 e^{- 0.14 t}$ as an equation.

$y = 60 e^{- 0.14 t}$

Step 2

Interchange the variables.

$t = 60 e^{- 0.14 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $60 e^{- 0.14 y} = t$ .

$60 e^{- 0.14 y} = t$

Step 3.2

Divide each term in $60 e^{- 0.14 y} = t$ by $60$ and simplify.

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Step 3.2.1

Divide each term in $60 e^{- 0.14 y} = t$ by $60$ .

$\frac{60 e^{- 0.14 y}}{60} = \frac{t}{60}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $60$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{60 e^{- 0.14 y}}{60} = \frac{t}{60}$

Step 3.2.2.1.2

Divide $e^{- 0.14 y}$ by $1$ .

$e^{- 0.14 y} = \frac{t}{60}$

Step 3.3

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

$\ln (e^{- 0.14 y}) = \ln (\frac{t}{60})$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{- 0.14 y})$ by moving $- 0.14 y$ outside the logarithm.

$- 0.14 y \ln (e) = \ln (\frac{t}{60})$

Step 3.4.2

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

$- 0.14 y \cdot 1 = \ln (\frac{t}{60})$

Step 3.4.3

Multiply $- 0.14$ by $1$ .

$- 0.14 y = \ln (\frac{t}{60})$

Step 3.5

Divide each term in $- 0.14 y = \ln (\frac{t}{60})$ by $- 0.14$ and simplify.

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Step 3.5.1

Divide each term in $- 0.14 y = \ln (\frac{t}{60})$ by $- 0.14$ .

$\frac{- 0.14 y}{- 0.14} = \frac{\ln (\frac{t}{60})}{- 0.14}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- 0.14$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{- 0.14 y}{- 0.14} = \frac{\ln (\frac{t}{60})}{- 0.14}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{t}{60})}{- 0.14}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

$y = - \frac{\ln (\frac{t}{60})}{0.14}$

Step 3.5.3.2

Multiply by $1$ .

$y = - \frac{1 \ln (\frac{t}{60})}{0.14}$

Step 3.5.3.3

Factor $0.14$ out of $0.14$ .

$y = - \frac{1 \ln (\frac{t}{60})}{0.14 (1)}$

Step 3.5.3.4

Separate fractions.

$y = - (\frac{1}{0.14} \cdot \frac{\ln (\frac{t}{60})}{1})$

Step 3.5.3.5

Divide $1$ by $0.14$ .

$y = - (7.\overline{142857} \frac{\ln (\frac{t}{60})}{1})$

Step 3.5.3.6

Divide $\ln (\frac{t}{60})$ by $1$ .

$y = - (7.\overline{142857} \ln (\frac{t}{60}))$

Step 3.5.3.7

Multiply $7.\overline{142857}$ by $- 1$ .

$y = - 7.\overline{142857} \ln (\frac{t}{60})$

Step 4

Replace $y$ with $C^{- 1} (t)$ to show the final answer.

$C^{- 1} (t) = - 7.\overline{142857} \ln (\frac{t}{60})$

Step 5

Verify if $C^{- 1} (t) = - 7.\overline{142857} \ln (\frac{t}{60})$ is the inverse of $C (t) = 60 e^{- 0.14 t}$ .

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Step 5.1

To verify the inverse, check if $C^{- 1} (C (t)) = t$ and $C (C^{- 1} (t)) = t$ .

Step 5.2

Evaluate $C^{- 1} (C (t))$ .

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Step 5.2.1

Set up the composite result function.

$C^{- 1} (C (t))$

Step 5.2.2

Evaluate $C^{- 1} (60 e^{- 0.14 t})$ by substituting in the value of $C$ into $C^{- 1}$ .

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} \ln (\frac{60 e^{- 0.14 t}}{60})$

Step 5.2.3

Cancel the common factor of $60$ .

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Step 5.2.3.1

Cancel the common factor.

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} \ln (\frac{60 e^{- 0.14 t}}{60})$

Step 5.2.3.2

Divide $e^{- 0.14 t}$ by $1$ .

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} \ln (e^{- 0.14 t})$

Step 5.2.4

Use logarithm rules to move $- 0.14 t$ out of the exponent.

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} (- 0.14 t \ln (e))$

Step 5.2.5

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

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} (- 0.14 t \cdot 1)$

Step 5.2.6

Multiply $- 0.14$ by $1$ .

$C^{- 1} (60 e^{- 0.14 t}) = - 7.\overline{142857} (- 0.14 t)$

Step 5.2.7

Multiply $- 7.\overline{142857} (- 0.14 t)$ .

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Step 5.2.7.1

Multiply $- 0.14$ by $- 7.\overline{142857}$ .

$C^{- 1} (60 e^{- 0.14 t}) = 1 t$

Step 5.2.7.2

Multiply $t$ by $1$ .

$C^{- 1} (60 e^{- 0.14 t}) = t$

Step 5.3

Evaluate $C (C^{- 1} (t))$ .

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Step 5.3.1

Set up the composite result function.

$C (C^{- 1} (t))$

Step 5.3.2

Evaluate $C (- 7.\overline{142857} \ln (\frac{t}{60}))$ by substituting in the value of $C^{- 1}$ into $C$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- 7.\overline{142857} \ln (\frac{t}{60}))}$

Step 5.3.3

Simplify $- 7.\overline{142857} \ln (\frac{t}{60})$ by moving $7.\overline{142857}$ inside the logarithm.

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln ({(\frac{t}{60})}^{7.\overline{142857}}))}$

Step 5.3.4

Apply the product rule to $\frac{t}{60}$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (\frac{t^{7.\overline{142857}}}{60^{7.\overline{142857}}}))}$

Step 5.3.5

Raise $60$ to the power of $7.\overline{142857}$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (\frac{t^{7.\overline{142857}}}{5024355493192.33}))}$

Step 5.3.6

Multiply by $1$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (\frac{1 t^{7.\overline{142857}}}{5024355493192.33}))}$

Step 5.3.7

Factor $5024355493192.33$ out of $5024355493192.33$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (\frac{1 t^{7.\overline{142857}}}{5024355493192.33 (1)}))}$

Step 5.3.8

Separate fractions.

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (\frac{1}{5024355493192.33} \cdot \frac{t^{7.\overline{142857}}}{1}))}$

Step 5.3.9

Divide $1$ by $5024355493192.33$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (0 (\frac{t^{7.\overline{142857}}}{1})))}$

Step 5.3.10

Divide $t^{7.\overline{142857}}$ by $1$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{- 0.14 (- \ln (0 t^{7.\overline{142857}}))}$

Step 5.3.11

Multiply $- 0.14 (- \ln (0 t^{7.\overline{142857}}))$ .

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Step 5.3.11.1

Multiply $- 1$ by $- 0.14$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{0.14 \ln (0 t^{7.\overline{142857}})}$

Step 5.3.11.2

Simplify $0.14 \ln (0 t^{7.\overline{142857}})$ by moving $0.14$ inside the logarithm.

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 e^{\ln ({(0 t^{7.\overline{142857}})}^{0.14})}$

Step 5.3.12

Exponentiation and log are inverse functions.

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 {(0 t^{7.\overline{142857}})}^{0.14}$

Step 5.3.13

Apply the product rule to $0 t^{7.\overline{142857}}$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 (0^{0.14} {(t^{7.\overline{142857}})}^{0.14})$

Step 5.3.14

Raise $0$ to the power of $0.14$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 (0.01\overline{6} {(t^{7.\overline{142857}})}^{0.14})$

Step 5.3.15

Multiply the exponents in ${(t^{7.\overline{142857}})}^{0.14}$ .

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Step 5.3.15.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 (0.01\overline{6} t^{7.\overline{142857} \cdot 0.14})$

Step 5.3.15.2

Multiply $7.\overline{142857}$ by $0.14$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 60 (0.01\overline{6} t)$

Step 5.3.16

Multiply $0.01\overline{6}$ by $60$ .

$C (- 7.\overline{142857} \ln (\frac{t}{60})) = 1 t$

Step 5.4

Since $C^{- 1} (C (t)) = t$ and $C (C^{- 1} (t)) = t$ , then $C^{- 1} (t) = - 7.\overline{142857} \ln (\frac{t}{60})$ is the inverse of $C (t) = 60 e^{- 0.14 t}$ .

$C^{- 1} (t) = - 7.\overline{142857} \ln (\frac{t}{60})$