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Calculus/Calc Homework

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Expert: - 9/20/2010


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(subscript 0)(1-e^(-t/a))

(the maximum charge capacity is Q(subscript 0) 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
I will refer to Q(subscript 0) as Q0.

a) If Q(t) = Q0(1-e^(-t/a)), then to find the inverse, solve for t.
That is, solve Q = Q0(1-e^(-t/a)) for t in terms of Q.

First divide both sides by Q0, then subtract 1 from both sides,
and then take the negative of both sides.

This should be f(Q) = e^(-t/a) for some function f(Q).
Taking the ln() of both sides gives ln(f(Q)) = -t/a.

Multiply both sides by -a and there is t in terms of Q.

b) Solve the equation (Q0)/2 = Q0(1 - e^(-t/2)).
That should reduce to e^(-t/2) = 1/2.
Once that point is reached, it can be said that e^(t/2) = 2.
After that, take the ln() of both sides, cancelling the exponential on the left.  To complete the process, multiply both sides by 2.

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