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Problem 29 Hard Difficulty

A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table.

(a) Make a scatter plot of the data.
(b) Use a graphing calculator to find an exponential curve $ f(t) = a \cdot b^t $ that models the bacteria population $ t $ hours later.
(c) Graph the model from part (b) together with the scatter plot in part (a). Use the TRACE feature to determine how long it takes for the bacteria count to double.

Answer

a) 37
b) $f(t)=(36.78) \cdot(1.07)^{t}$
(c) $x=20.727273$ $Y=139.13451$

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Video Transcript

all right in this problem, we're going to use a graphing calculator and take the data about the bacteria count over time and type it in to our statistics lists. So we pressed the stat button, and then we go into edit, and then we type the numbers into list one in less, too, and the first thing we want to do with it is make a scatter plot. So we need to go into the stat plot window, which is second y equals. Go into the menu for ah stat plot one and turn it on and make sure it's a scatter plot using list one and list, too. The next thing we can do is set a good window for our plot so we can go to zoom and then down to Zoom nine, which is IAM stat. So there's our scatter plot. We can see it's curving like an exponential growth function would, and the next thing we want to do is find the exponential model so we can go back to stat over to calculate and go down until we find exponential regression. Select that use list one and list, too, and then let's store the regression equation in our Y equals menu. So from here we go to the vars button over to why variables choose function and choose why one. And now we can calculate. So these are the values that would go into our exponential model. We have approximately y equals a r y equals 36.89 times, 1.7 to the X power. And if we press why equals will see that we did pace that equation in there with a whole lot of decimal places to increase the accuracy, the next thing we want to do is graph this model along with the scatter plot. So we compress graph and there we have that model graft, and now we're going to use the trace feature to determine how long it takes for the bacteria count to double. So I'm going to press trace, and then I'm going to use my arrow down button to make sure that my cursor is actually on the bottle and not on the scatter plot. Now let's pick a value for X. Let's say X is oh right about let's pick a value for why, for the bacteria counts so right here I have y equals 70 approximately, and I have X equals 10. So 10 hours, 70 bacteria 70. Whatever the units of count are for the bacteria. Now, let's continue tracing until we get to 140 bacteria, and that would be double. This is just a rough estimate for how long it takes to double. So we're getting closer to 1 40 Okay, so that's about as close as we're going to get to 1 40 we see that it is at about 20.7 hours. So that tells us that it took about 10.7 hours for the bacteria to double.