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Problem 89 Hard Difficulty

The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge $ Q $ remaining on the capacitor (measured in microcoulombs, $ \mu C $ ) at time $ t $ (measured in seconds).
(a) Use a graphing calculator or computer to find an exponential model for the charge.
(b) The derivative $ Q'(t) $ represents the electric current (measured in microamperes, $ \mu A $ ) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when $ t = 0.04 s. $ Compare with the result of Example 2.1.2.

Answer

a. $a \approx 100.0124369$
b. $\approx-670.63 \mu \mathbf{A}$

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Video Transcript

for this problem. We have a table of data and we're going to take it, take a graphing calculator and we're going to find the exponential model. So we go into the stat menu and then we go into the edit menu and we type the times into list one and the values of Q. The charge into list, too. And so our goal, then, is to find an exponential model for this. So we go back to stat and over to calculate, and then we go down until we find exponential regression. There it is. Press enter. We're using List one and list, too. We would like to store this equation in our Y equals menu. So when you get to store regression equation, you can go to variables over to why variables choose function and choose why one and then we're going to calculate. So here the values that we put into the exponential model and if you're going to write this down, you can write something roughly like this. Notice that the base is 4.5 times 10 to the negative fifth, so that can be a challenging one to write without putting a whole bunch of numbers in now what we're going to do is go back to the calculator and think about the derivatives. So it tells us in part B that the derivative represents the current and we're going to estimate the current when the time is 0.4 So we want to estimate the derivative, and that means estimate the slope of the tangent line and the calculator can help us with this. So what I'm going to do is graph the function it's already typed into my Y equals menu because I pasted it there earlier and we have a scatter plot turned on, and so we can go into Zoom Number nine Zoom stat. And there we see the scatter plot and we see the model plotted. So now for the Tangent Line Slope, we can go into the Draw menu, which is second program. Choose number five, draw a tangent and then type in 0.4 We're finding the tangent at the X value 0.4 That will be the electric current at that time. Press enter and you see the tangent line was drawn and at the bottom of the screen you see the value of the You see the whole equation of the tangent line here and the first number before acts. That's the slope. So it's approximately negative. 670.6 now, if you look back a chapter two at example, one in section 2.1. You'll see that the way they calculated the charge there they were a couple different ways the way they calculated the current. In one method, they got negative 670 and using another method, they got negative 674.75 So we're close to both of those answers. This is very consistent.