Exercise · ZW-EXPONENTIAL-GROWTH-AND-DECAY-004 · ★★☆ Intermediate
A substance has a half-life of 5 years and satisfies \(y'=ky\). What is \(k\)?
Choose one option.
Show answer
- D. \(k=-\tfrac{\ln2}{5}\)
Explanation
From \(e^{5k}=\tfrac12\) we get \(k=-\tfrac{\ln2}{5}\), negative because it is decay.
Review the concept: Exponential growth and decay
A quantity \(y\) with initial value \(y_0\) grows or decays exponentially when its rate of change is proportional to itself, with constant \(k\): it grows if \(k>0\), decays if \(k<0\).
\[y'=ky\;\Longrightarrow\; y(t)=y_0\,e^{kt}\]
More practice problems: exponential growth and decay
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-001 Solve y'=3y with y(0)=2.
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-002 Solve y'=-tfrac12 y with y(0)=4.
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-003 A population satisfies y'=ky, with y(0)=100 and y(2)=200. What is y(6)?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-005 A capital grows with y'=0.04y. After roughly how long does it double?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-006 In y'=ky with k<0 and y(0)>0, what happens to y(t) as t→∞?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-007 A 100 mg isotope keeps 80 mg after 3 years. With exponential decay, how many mg remain after 9 year…
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-008 With y'=0.1y, about when does the quantity triple?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-009 If y'=ky with y(1)=6 and y(3)=24, what is y(0)?
← ZW-EXPONENTIAL-GROWTH-AND-DECAY-003 ZW-EXPONENTIAL-GROWTH-AND-DECAY-005 →