When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $Q(t)=Q_{0}\left(1-e^{-t / a}\right)$ (The maximum charge capacity $Q_{0}$ is and t is measured in seconds.) How long does it take to recharge the capacitor to 90% of capacity if $a=5$ ?
A) $-5 \ln (1)$ seconds
B) $-5 \ln \left(\frac{1}{10}\right)$ seconds
C) $-\ln \left(\frac{4}{9}\right)$ seconds
D) $-5 \ln (10)$ seconds
E) $-5 \ln \left(\frac{9}{10}\right)$ seconds

Question

Quizplus · Accepted Answer

When $Q(t)=0.9Q_{0}$, $$0.9Q_{0}=Q_{0}\left(1-e^{-t / a}\right)$$$$0.9=1-e^{-t / a}$$$$e^{-t / a}=0.1$$$$-t / a=\ln (0.1)$$$$t=-a \ln (0.1)=5 \ln (10)$$

When a Camera Flash Goes Off, the Batteries Immediately Begin Q(t)=Q0(1−e−t/a)Q(t)=Q_{0}\left(1-e^{-t / a}\right)Q(t)=Q0​(1−e−t/a)

When a Camera Flash Goes Off, the Batteries Immediately Begin $Q(t)=Q_{0}\left(1-e^{-t / a}\right)$