Chemists commonly create a solution by mixing two products of differing concentrations together. For example, a chemist could have large amounts of a $\sage {concA}\%$ acid solution and a $\sage {concB}\%$ acid solution, but need a $\sage {totalVolume}$ liter $\sage {concTotal}$% solution. Construct a model that describes the amount of acid in a $\sage {concTotal}\%$ acid solution, $A_{\sage {concTotal}}$, in terms of the volume of the $\sage {concA}\%$ acid solution, $v$.

$A_{\sage {concTotal}} = \answer {\sage {eqPartD6}}$

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.

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

A company sells doughnuts. They incur a fixed cost of $$\sage {fixedCost1}$ for rent, insurance, and other expenses. It costs $$\sage {productionCost1}$ to produce each doughnut. The company sells each doughnut for $$\sage {sellingPoint1}$. Construct a model that describes their total revenue, $R$, as a function of the number of doughnuts, $x$, they produce.

$R(x) = \answer {\sage {revenue1}}$

Kepler’s Third Law: The square of the time, $T$, required for a planet to orbit the Sun is directly proportional to the cube of the mean distance, $a$, that the planet is from the Sun. Assume that Mars’ mean distance from the Sun is $\sage {a4}$ AUs and it takes Mars about $\sage {T4}$ months to orbit the Sun. Write the equation that describes time $T$ (years) in terms of the mean distance, $a$ (AUs).

$T(a) = \answer [tolerance=0.05]{\sage {k4}} a^{\answer {3/2}}$

The half-life of carbon-14 is 5,730 years. Describe the age in years of an object in terms of the ratio of carbon-14, $r = \frac {C}{C_0}$, remaining.

$t(r) = \answer [tolerance=0.01]{\sage {1/k5}} \ln (\answer {r})$

Two UFPD are patrolling the campus on foot. To cover more ground, they split up and begin walking in different directions. Office A is walking at $\sage {officerAspeed3}$ mph while Office B is walking at $\sage {officerBspeed3}$ mph. Construct a model that describes their total distance from each other, $T_2$, as a function of minutes, $m$, that have passed if they were walking in exactly 90 degrees from each other (e.g., North/East).

$T_2(m) = \answer {\sage {partC3}}$

A population of bacteria $\sage {rate8}$ every hours. If the culture started with $\sage {initialBacteria8}$, write the equation that models the bacteria population after $t$ hours.

$P(t) = \answer {\sage {initialBacteria8}} \answer {\sage {choose8+2}}^{\answer {t}}$