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Problem 62 Hard Difficulty

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by
$$ Q(t) = Q_0 (1 - e^{\frac{-t}{a}}) $$
(The maximum charge capacity is $ Q_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

a) $t=-a \ln \left[1-\frac{Q}{Q_{o}}\right]$
b) $\approx 4.605$ seconds

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Video Transcript

all right. So here we have our equation. And in this equation, T is the input, and two is the output. So for the inverse, we would switch those and Q would be the input and t would be the output. And that means the meaning of the inverse is that we're going to find time as a function of charge, whereas what we have right now is charge as a function of time. Typically, when we find an inverse, we start by switching X and Y and then solving for why, in this case, that would mean switching Q and T and then solving for T. I do want to sell for tea, but I'm going to be a little bit rebellious. And I'm not going to switch Q and T because I want T to stand for time and cute to stand for charge. And if I switch them, their meaning is going to switch. So I'm just going to isolate E. So start by dividing both sides by Q not and then subtracting one from both sides that should say minus and then multiplying both sides by negative one. That's going to give us one minus. Q. over Q not equals E to the negative. T over a and then we'll take the natural log of both sides to have natural log of one minus. Q. Over Q. Not equals negative. T over a and then to get to you by itself will multiply both sides by the opposite of a so we have t equals the opposite of a times the natural log of one minus. Q. Over. Q. Not okay, so we'll count. That is our inverse time as a function of charge. Now we want to use that to figure out how long it takes to recharge the capacitor to 90% of capacity of a equals two. So we want to find the time, and we know that Q is 90%. And we know that a is to now when we say that Q is 90% we mean it's 90% of Q. Not so Q equals 900.9 times. Q. Not now our function was t equals negative. A so negative two times the natural log of one minus que over Q Not, which would be 0.9 Q. Not over. Q. Not we can simplify that. That's t equals negative two times a natural log of one minus 10.9. Let's keep simplifying. T equals negative two times the natural log of 20.1 and let's go ahead and approximate that we can put that in a calculator.

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