Question

Suppose the following graph represents the number of bacteria in a culture t hours after the start of an experiment. a. At approximately what time is the instantaneous growth rate the greatest, for (StS 36? Estimate the growth rate at this time. b. At approximately what time in the interval Osts 36 is the instantaneous growth rate the least? Estimate the instantaneous growth rate at this time. c. What is the average growth rate over the interval Osts 36? a. The instantaneous rate of growth is the greatest at approximately t = | (Round to the nearest whole number as needed. Use a comma to separate answers as needed.) The growth rate at this time(s) is approximately (Round to the nearest whole number as needed.) b. The instantaneous rate of growth is the least at approximately t= (Round to the nearest whole number as needed. Use a comma to separate answers as needed.) The growth rate at this time(s) is approximately (Round to the nearest whole number as needed.) Enter your answer in each of the answer boxes.

Answer 1

If we take a straight line and start moving it along the curve shown, we get the tangents at the different points.

a) The instantaneous growth rate is highest at the point where the slope of the tangent line is highest. So moving the straight line we can predict that that slope is highest at approximatley t=20 .

To find the growth rate we find the approximate values of number of bacteria at t=20 and Number of bacteria at t=21,

At t=20, Number of bacteria =3250 and

At t=21, Number of bacteria =3500

So that the growth rate is $3500-3250=250$

Hence, growth rate is highest at $t=20$ and its equal to 250

b) The instantaneous growth rate is least at the point where the slope of the tangent line is least. So moving the straight line we can predict that that slope is least at approximatley t=0.

To find the growth rate we find the approximate values of number of bacteria at t=20 and Number of bacteria at t=21,

At t=0, Number of bacteria =400 and

At t=6, Number of bacteria =500

So that the growth rate is ${ 500-400 \over 6-0}\approx 16.67$

Hence, growth rate is highest at $t=0$ and its equal to 17

c) At the end points we have

At t=0, Number of bacteria =400 and

At t=36, Number of bacteria =5000

Hence average growth rate is ${ 5000-400 \over 36-0}\approx 127.78$

That is average growth rate over the interval is 128