AP Calculus AB: Exponential Growth and Decay Models — Drill 1

About This Drill

AP Calculus AB: Exponential Growth and Decay Models — Drill 1 is a Math practice drill covering Exponential Growth and Decay Models. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice applying the differential equation \( \dfrac{dy}{dt} = ky \) and its solution \( y = Ce^{kt} \) to model exponential growth and decay, including half-life, doubling time, and interpreting the constant k in context. These skills appear on the AP Calculus AB exam in both multiple-choice and free-response questions.

Questions in This Drill

1. A quantity \( y \) satisfies \( \dfrac{dy}{dt} = -0.3y \). If \( y(0) = 50 \), which of the following gives \( y(t) \)?
2. A population grows according to \( \dfrac{dP}{dt} = 0.04P \). Approximately how long does it take for the population to double?
3. A radioactive substance decays according to \( A(t) = A_0 e^{-0.0231t} \), where \( t \) is in years. What is the half-life of the substance, to the nearest year?
4. A cup of coffee cools according to Newton's Law of Cooling: \( T(t) = 22 + 68e^{-0.05t} \), where \( T \) is temperature in °C and \( t \) is time in minutes. Which of the following best interprets the value \( k = -0.05 \)?
5. At time \( t = 0 \), a bacterial culture contains \( 100 \) bacteria. At time \( t = 2 \) hours, it contains \( 900 \) bacteria. Assuming exponential growth \( y = Ce^{kt} \), what is the approximate value of \( k \)?