When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by q=q(t)=k(1−e−t/a).

Question

(The maximum charge capacity is k and t is measured in seconds).

(a) Find the inverse of this function. (Explain its meaning.)

(b) How long does it take to recharge the capacitor to  percent of capacity if ?

Answer 1

a) I believe the equation for q(t) should be written as

q(t) = k(1 - e^-t/a)

To find the inverse, t(q), proceed as follows:

q = k(1 - e^-t/a)

q/k = 1 - e^-t/a

e^-t/a = 1 - q/k

ln(e^-t/a) = ln(1 - q/k)

ln( 1 - q/k ) = (-t/a) ln(e) = -t/a (since ln(e) = 1)

-t/a = ln( 1 - q/k )

t = -a ⋅ ln( 1 - q/k )

t(q) = -a ⋅ ln( 1 - q/k )

This gives the time required to reach a charge value of q.

b) The question is incomplete.