Objective 3 - Construct Log/Exp Models

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

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

Videos:

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.

$X(t) = \answer {\sage {initialAmount2}} \answer {e}^{\answer [tolerance=0.01]{\sage {k2}} t }$

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.

$C(t) = C_0 * \answer {e}^{\answer [tolerance=0.01]{\sage {k3}} t}$

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

$t(r) = \answer [tolerance=0.01]{\sage {1/k3}} \ln (\answer {r})$

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?

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