Advanced Math: Exponential Functions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A savings account starts with an initial deposit of $\$500$. The account earns $3%$ interest, compounded annually. Which function $A(t)$ represents the amount of money in the account after $t$ years?
   • A) $A(t) = 500(0.03)^t$
   • B) $A(t) = 500(1.03)^t$
   • C) $A(t) = 500(1.30)^t$
   • D) $A(t) = 500 + 1.03t$
2. The population of a small town decreases by $2%$ every year. If the current population is $15,000$, which expression models the population after $t$ years?
   • A) $15000(0.98)^t$
   • B) $15000(0.02)^t$
   • C) $15000(1.02)^t$
   • D) $15000 - 0.02t$
3. A painting was purchased for $\$1,200$. Its value $V(t)$, in dollars, $t$ years after the purchase is modeled by $V(t) = 1200(1.15)^t$. By what percentage does the value of the painting increase each year? (Grid-in)
4. The function $P(x) = 250(0.88)^x$ models the number of pests in a garden $x$ weeks after a treatment is applied. What is the best interpretation of the number $0.88$ in this context?
   • A) The number of pests decreases by $88%$ each week.
   • B) The number of pests remaining is $88%$ of the amount from the previous week.
   • C) The number of pests decreases by $0.88$ each week.
   • D) The treatment was $88%$ effective overall.
5. A rare coin is currently worth $\$400$. If its value triples every decade, which equation represents the coin's value $V(d)$ after $d$ decades?
   • A) $V(d) = 400(3)^d$
   • B) $V(d) = 400(\frac{1}{3})^d$
   • C) $V(d) = 3(400)^d$
   • D) $V(d) = 400 + 3^d$

Level 2: Medium (Standard SAT)

1. The value of a stock portfolio $V$, in dollars, is modeled by the equation $V(t) = 5000(1.045)^{t/12}$, where $t$ is the number of months since the portfolio was opened. What is the percentage increase in the portfolio's value exactly one year after it was opened? (Grid-in)
2. A radioactive isotope has a half-life of $8$ days. If a scientist begins with $200$ grams of the isotope, which function $M(d)$ models the mass of the isotope remaining after $d$ days?
   • A) $M(d) = 200(0.5)^{8d}$
   • B) $M(d) = 200(0.5)^{d/8}$
   • C) $M(d) = 200(8)^{d/0.5}$
   • D) $M(d) = 8(0.5)^{d/200}$
3. A car depreciates in value by $15%$ every year. If the original cost of the car was $\$24,000$, which of the following represents the car's value $V(t)$ after $t$ months?
   • A) $V(t) = 24000(0.85)^{12t}$
   • B) $V(t) = 24000(0.85)^{t/12}$
   • C) $V(t) = 24000(1.15)^{t/12}$
   • D) $V(t) = 24000(0.15)^{t/12}$
4. The function $h(x) = 150(2.4)^{x}$ models the number of users on an app $x$ weeks after its launch. The number of users increases by $p%$ each week. What is the value of $p$? (Grid-in)
5. The expression $12(1.08)^t$ models the height, in centimeters, of a sapling $t$ weeks after it is planted. If the expression is rewritten in the form $a(b)^{t/7}$ to model the height after $d$ days, what is the value of $b$?
   • A) $\frac{1.08}{7}$
   • B) $1.08$
   • C) $(1.08)^7$
   • D) $\sqrt{1.08}$

Level 3: Hard (Advanced SAT)

1. The population of a city is given by the function $P(t) = 45000(1.21)^{t/10}$, where $t$ is the number of years since 2000. Which of the following is the best interpretation of the number $1.21$ in this context?
   • A) The population increases by $21%$ every year.
   • B) The population increases by $121%$ every $10$ years.
   • C) The population increases by $21%$ every $10$ years.
   • D) The population multiplies by $1.21$ every year.
2. A colony of insects has an initial population of $N_0$. The population is observed to triple every $k$ hours. Which of the following equations gives the population $P(h)$ after $h$ hours?
   • A) $P(h) = N_0(3)^{h/k}$
   • B) $P(h) = N_0(3)^{k/h}$
   • C) $P(h) = N_0(\frac{1}{3})^{h/k}$
   • D) $P(h) = 3N_0(k)^{h}$
3. A biologist models the growth of a fungus using $A(t) = 50(2)^{3t}$, where $t$ is the time in days. The biologist wants to rewrite the function in the form $A(t) = a(b)^t$. What is the value of $b$? (Grid-in)
4. The function $f(x) = 100(0.81)^x$ can be rewritten in the form $f(x) = 100(b)^{2x}$, where $b$ is a constant. What is the value of $b$? (Grid-in)
5. A company's revenue $R$, in millions of dollars, $t$ years after $2010$ is modeled by $R(t) = 14(1.06)^t$. The company decides to create a new model $S(m)$ where $m$ is the number of months after $2010$. Which equation is the correct model for $S(m)$?
   • A) $S(m) = 14(1.06)^{m/12}$
   • B) $S(m) = 14(1.06)^{12m}$
   • C) $S(m) = 14(\frac{1.06}{12})^m$
   • D) $S(m) = \frac{14}{12}(1.06)^{m/12}$

Level 4: Very Hard (Slightly beyond SAT)

1. A scientist models the decay of a substance with the function $D(t) = 800(0.5)^{t/6}$, where $t$ is the time in hours. The scientist wants to rewrite the model to show the percentage of the substance that decays each hour. Which of the following expressions represents the percentage that decays per hour?
   • A) $100(1 - (0.5)^{1/6})$
   • B) $100(0.5)^{1/6}$
   • C) $100(1 - (0.5)^6)$
   • D) $\frac{50}{6}$
2. In the function $P(x) = a(b)^x$, $P(2) = 144$ and $P(4) = 324$. Assuming $b > 0$, what is the value of the initial amount $a$? (Grid-in)
3. The population of Town A is modeled by $P_A(t) = 2000(1.4)^t$ and the population of Town B is modeled by $P_B(t) = 3000(1.2)^t$, where $t$ is measured in years. In the model for the ratio of the population of Town A to Town B, $R(t) = \frac{P_A(t)}{P_B(t)}$, the ratio grows exponentially by $q%$ per year. What is the value of $q$ to the nearest tenth? (Grid-in)
4. A bank account earns a certain interest rate compounded annually. After $5$ years, the account balance has grown by exactly $30%$ of its original value. Which expression represents the annual interest rate as a decimal?
   • A) $0.30/5$
   • B) $1.30^{1/5}$
   • C) $1.30^{1/5} - 1$
   • D) $1 - 0.30^{1/5}$
5. If $f(x) = 10(2^x)$ and $g(x) = f(x + 3)$, what is the ratio of $g(x)$ to $f(x)$ for any real value of $x$? (Grid-in)