Objective 3 - Construct Log/Exp Models

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Link to textbook: Construct a model equation for the real-life situation.

You can print out these notes to follow along with the video below and keep notes to organize your thoughts.

A population of bacteria $\sage {rate1}$ every hours. If the culture started with $\sage {initialBacteria1}$ , write the equation that models the bacteria population after $t$ hours.

$P(t) = \answer {\sage {initialBacteria1}} \answer {\sage {choose1+2}}^{\answer {t}}$

There is initially $\sage {initialAmount2}$ grams of element $X$ . The half-life of element $X$ is $\sage {halfLifeYears2}$ years. Describe the amount of element $X$ remaining as a function of time, $t$ , in years. Use $e$ as the base of your exponential model and exact values.

$X(t) = \answer {\sage {initialAmount2}} e^{\answer {\sage {k2}t} }$

The exponent’s coefficient is not $\frac {1}{\sage {halfLifeYears2}}$ nor is it $\frac {-1}{\sage {halfLifeYears2}}$ . To find the correct coefficient, take the base exponential equation $\text {current amount } = \text { initial amount } e^{ct}$ , use what you know about the half-life time and amount, then solve the equation for $c$ . If you are having trouble, refer back to the objective on ”Solving exponential equations”. Do not use a calculator - use exact values to get a correct answer.

The half-life of carbon-14 is 5,730 years.

Part A. Describe the amount of carbon-14 remaining after $t$ years. The initial amount of carbon-14, $C_0$ , is already included below.

$C = C_0 * \answer {e}^{\answer {\sage {k3}t} }$

Part B. Solve the equation above for $t$ written in terms of the ratio of carbon-14 remaining, $r = \frac {C}{C_0}$ .

$t = \answer [tolerance=0.001]{\sage {1/k3}} \ln (\answer {r})$

Part C. The equation above is used to carbon-date objects. To solidify this idea, use the model in Part B. to solve the following problem.

A bone fragment is found that contains $\sage {percent3}\%$ of its original carbon-14. To the nearest year, how old is the bone?

$\answer {\sage {old3}} \, \text { years old}$

Part A. The exponent’s coefficient is not $\frac {1}{5730}$ nor is it $\frac {-1}{5730}$ . To find the correct coefficient, take the base exponential equation $\text {current amount } = \text { initial amount } e^{ct}$ , use what you know about the half-life time and amount, then solve the equation for $c$ . If you are having trouble, refer back to the objective on ”Solving exponential equations”. Do not use a calculator - use exact values to get a correct answer.

Part B. Now that you found the exponent’s coefficient, you have modeled the general equation as $C = C_0 e^{ct}$ . Since it asks you to rewrite the equation in terms of $r = \frac {C}{C_0}$ , start by converting the equation into one with $r$ instead of $C$ and $C_0$ . Then, solve the equation for $t$ .

Part C. What does $r = \frac {C}{C_0}$ represent? If we know this, then we could use the equation you built in Part B. to solve for the age of the bone.

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