1. An investment is modeled by the function $f(t) = 2500(1.08)^t$, where $t$ is in years. If this is rewritten to show monthly growth as $f(t) = 2500(b)^{12t}$, what is the monthly growth factor $b$? Round to four decimal places. $b$ = ___
2. An exponential growth function is given by $P(t) = P_0(b)^t$, where $b > 1$ is the annual growth factor. If this function is rewritten as $P(t) = P_0(r)^{4t}$ to represent quarterly growth, what is the relationship between $r$ and $b$?
3. The amount of a decaying substance is modeled by $A(t) = 800(0.92)^t$, with $t$ in years. If the function is rewritten to show quarterly decay as $A(t) = 800(k)^{4t}$, what is the quarterly decay factor $k$? Round to four decimal places. $k$ = ___
4. A bacterial population is modeled by $P(t) = 300(1.2)^{\frac{t}{3}}$, where $t$ is in hours. Rewrite this function in the standard form $P(t) = 300(b)^t$ to find the hourly growth factor. What is $b$? Round to four decimal places. $b$ = ___
5. A car's value is modeled by $V(t) = 20000(0.84)^t$, where $t$ is in years. Which function below correctly represents the value decaying semiannually (twice per year)?
6. A town's population grows at $8\%$ per year: $P(t) = 20{,}000(1.08)^t$. The equivalent monthly growth factor is $\bigl((1.08)^{1/12}\bigr)$. The effective monthly growth rate (as a percent, rounded to three decimal places) is ___%.
7. A machine depreciates at $20\%$ per year: $V(t) = 30{,}000(0.80)^t$. Which expression correctly rewrites this model to show the equivalent **quarterly** depreciation factor?
8. A bacterial culture triples each hour: $B(t) = 200(3)^t$, where $t$ is in hours. Rewritten for minutes ($k = 60$), the per-minute growth factor is $3^{1/60}$. The effective per-minute growth rate (rounded to two decimal places) is ___%.
9. A laptop depreciates at $25\%$ per year. Its value model is $V(t) = 1{,}200(0.75)^t$. What is the approximate effective **quarterly** decay rate?
10. A population model is given as $P(t) = 10{,}000(1.12)^t$ with $t$ in years. Which expression gives the equivalent model with $t$ measured in **months**?