Inverse function Word Problem

Question

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)(1-e^(-t/a)) where Q_0 is the maximum charge capacity and t is measured in seconds.

 

a). Find the inverse of this function and explain its meaning.

b). How long does it take to recharge the capacitor to 90% of capacity if a=2

Answer 1

Hi Max, I'm not sure if you still need help since your question was posted quite some time back. Anyway, hope the following helps.

 

Concepts that you need to know:

1) The function given is Q(t), which means Q is given in terms of t. Inverse of the function needs you to give t in terms of Q, which is the reverse way. So, your aim is to make t the subject of the given formula.

2) When e^x = y, x = ln y where y > 0.

3) 90% = 0.9, maximum capacity is Q₀. I believe (b) is referring to charging it to 90% of MAXIMUM capacity. So, this means that the question wants you to solve for t when Q(t) = 0.9Q₀.

 

Solution:

(a)

 

Q = Q₀(1 - e^-t/a)

Q/Q₀ = 1 - e^-t/a

e^-t/a = 1 - Q/Q₀

-t/a = ln(1 - Q/Q₀)

t = (-a) ln(1 - Q/Q₀)

t = (-a) ln(1 - Q/Q₀) tells us the time taken in seconds to recharge the flash's capacitor till it stores electric charge Q after the camera flash goes off.

 

(b)

Note: Since the question did not restrict us to using answer from (a) to do (b), we can use the given formula Q = Q₀(1 - e^-t/a) to solve part (b). You may prefer to do it in this manner, just in case your answer to part (a) is wrong. Alternatively, we can use answer from part (a), which will be a more direct method since we are trying to find the time taken for the recharging process. Anyway, both works fine by substituting Q = 0.9Q₀ and a = 2. You can use both to check your answer.

I will use part (a) answer to solve this.

 

t = (-a) ln(1 - Q/Q₀)

  = (-2) ln(1 - 0.9Q₀/Q₀)

  = (-2) ln(0.1)

  = 4.61 (rounded off to 3 s.f.)

In conclusion, it takes 4.61 s to recharge the capacitor to 90% of capacity if a=2.