When a camera flash goes off, the batteries immediately begin to recharge the fl

Question

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by the following.

Q(t) = Q0(1 ? e?t/a)

(The maximum charge capacity is Q0 and t is measured in seconds.)

(a) Find the inverse of this function.

Explain its meaning.

1)This gives us the charge Q obtained within a given time t.

2)This gives us the time t necessary to obtain a given charge Q.    

3)This gives us the time t with respect to the maximum charge capacity Q0.

(b) How long does it take to recharge the capacitor to 77% of capacity if a = 5? (Round your answer to one decimal place.) in sec

Explanation / Answer

let y = Qo(1 - e^(-t/a))

switch t and y for the inverse, then solve for y
t = Qo(1 - e^(-y/a))

t/Qo = 1 - e^(-y/a)
1 - t/Qo = e^(-y/a)
ln(1 - t/Qo) = ln(e^(-y/a)) = -y/a
y = -a*ln(1 - t/Qo) <===== inverse

the original function was charge as a function of time...
the inverse will be time to reach a certain charge
In this case, it makes more sense to maintain the sense of the original variables, so let's not switch t and Q...

t = -a*ln(1 - Q/Qo) <==== inverse with variables maintained

b) for 90% of original charge, Q / Qo = .9
since a = 2
t = -2ln(1 - .9) = -2ln(.1) = 4.6 seconds to recharge to 90%

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