Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

A common inhabitant of human intestines is the bacteriumEscherichia coli. A cell of this bacterium in a nutrient-brothmedium divides into two cells every 20 minutes. The initialpopulation of a culture is 60 cells.(a) Find the relative growth rate.(b) Find an expression for the number of cells after $t$ hours.(c) Find the number of cells after 8 hours.(d) Find the rate of growth after 8 hours.(e) When will the population reach $20,000$ cells?

a) $$\ln (8)$$b) $$y(t)=60 \times 8^{t}$$c) $$1,006,632,960$$d) $$209.32~billion ~cells ~per ~hour$$e)$$ 2.79 ~hours$$

Calculus 1 / AB

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

Derivatives

Differentiation

Applications of the Derivative

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

02:18

A common inhabitant of hum…

04:49

$$\begin{array}{l}{\te…

07:57

Bacteria population A bact…

08:01

A bacteria culture grows w…

08:36

A cell of the bacterium $E…

08:41

04:10

A bacteria culture initial…

06:14

04:20

Growth of Bacteria The num…

07:00

05:59

Bacteria Culture The count…

10:44

Population of a Bacteria C…

02:37

Bacteria Growth The number…

05:26

01:43

E. coli Growth A strain of…

04:38

A culture of bacteria in a…

00:44

03:33

A bacteria culture starts …

09:43

Hey guys, welcome back for this problem. You know, the initial population is 60 and that it doubles every 20 minutes. Say that why subzero is equal to 60 Newman say, at 20 minutes or 1/3 of an hour wise eagle toe 1 20 We'll keep this all in hours. We'll see why of 1/3 is equal to 100 in 20. Because it's exponential growth. We will have our equation. Do you? Why over DT does he go to K y and then upon integrating VR theorem in the textbook that comes out to Why Sebti to the initial y value? Why subzero be Katie in part a were asked to solve for K. Now it's sulphuric, So let's look at a 20 minute time period or went 1/3 of an hour has passed and there are 1 20 Let's plug those values. So why City would be 1 20 for after 1/3 of an hour has passed, you know that our initial Y value is 60 e to the K. Like we said, 1/3 of an hour has passed to invite each side by 60 with two to go to the E to the 1/3 k we can take the natural log of both sides. We had natural log of to people to 1/3 k All right, kay people to three natural log of to. She's also go to the natural law two. Cubed just equal to the natural log of eight for part a k. Neil too natural log of it Now in part B, we were asked to find the any given time what the formula will be. It's basically were asked to just figure out this formula right here and put him some constants. So and b, we now have that. Why city At any time, t it's hell to Weiss, a bow, which is 60 e raised to the Ellen eight. Because as r k value times any time t now we can actually rewrite that as 60 times e to the Ellen A. Raised the T based on the property that natural logarithms that point the E and L, in turn, will cancel out who left with 60 times eight raised to the T so it any time t are y value will be equal to 60 times eight. Race to the team for part C. of this problem Rest if I and how many cells are there? AT T is equal to eight hours. So at T is about eight hours. We have Why some eight 0 to 60 times eight to the It would be a huge number and we end up with one billion six million 600 32,000 900 and 60. That is the amount of sales after eight hours. Now we want to find the growth rate. After eight hours, we'll go back to this equation over here for a growth rate equation to a party we will write. D Why DT que y Okay. We know is natural log of eight. And in our multiply it by r y value just found out. Is this guy so plugged in? One million is 1,000,000,006 million 600 and 32,000 9 60 and we find that our growth rate equals 209 1,000,000,000 cells per hour. That is our growth rate. De Y t t. There's one more part of this problem now as to find when the population will reach 20,000 cells. We want to go back to this guy, right? here. Our answer to Part B. We want to plug in 20,000 for Y city. 20 1000 is equal to 60 times eight raise to the T. We can divide both sides by 60. And when we do, we get 333 0.3. Field eight, race the tea. We now take the natural log of both sides. Ln 333 0.3 Siegel to the l N eight. Race the tea based on the properties of natural longer them on this right hand side, Tekin, come outfront are left with tea, Ellen to the eight. That would say that T is able to Ln 333.3 divided by Ellen of eight. And when you that math it comes out to point 79 20,000 cells, it will take two point 79 hours. And that is our answer to party. And that includes all five answer parts. Thanks for watching

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

A common inhabitant of human intestines is the bacterium Escherichia coli, n…

$$\begin{array}{l}{\text { E. coli population A common inhabitant of hum…

Bacteria population A bacteria culture grows with constantrelative growt…

A bacteria culture grows with constant relative growth rate. The bacteria co…

A cell of the bacterium $E$ coli divides into two cells every 20minutes …

A bacteria culture grows with constant relative growth rate.The bacteria…

A bacteria culture initially contains 100 cells and grows at a rate proporti…

A bacteria culture initially contains 100 cells and grows at arate propo…

Growth of Bacteria The number $N$ of bacteria present in a culture at time $…

Bacteria Culture The count in a culture of bacteria was 400 after 2 hours an…

Population of a Bacteria Culture The logistic growth model$$P(t)=\fr…

Bacteria Growth The number of bacteria in a petri dish culture after $t$ hou…

Bacteria Growth The number of bacteria in a culture is increasing according …

E. coli Growth A strain of $E$ coli Beu 397 -recA441 is placed into a nutrie…

A culture of bacteria in a Petri dish has an initial population of 1500 cell…

Bacteria Growth The number $B$ of bacteria in a petridish culture after $t$ …

A bacteria culture starts with 500 bacteria and doubles in size every half h…

03:47

The edge of a cube was found to be 30 $\mathrm{cm}$ with a possible error in…

02:20

Find the derivative of the function. Simplify where possible.$h(t)=\cot …

00:18

A function is given by a table of values, a graph, aformula, or a verbal…

01:18

Find the numerical value of each expression.(a) $\sinh (\ln 2)$(b) $…

02:52

Zoom in toward the points $(1,0),(0,1),$ and $(-1,0)$ on the graph of the fu…

01:09

The graph of $f$ is given. State, with reasons, the numbers at which $f$ is …

01:41

Verify the given linear approximation at $a=0 .$ Then determine the values o…

01:19

The equation of motion of a particle is $s=2 t^{3}-7 t^{2}+4 t+1,$ where $s$…

04:19

Let $$f(x)=\sqrt[3]{x}$$(a) If $a \neq 0,$ use Equation 2.1 .5 to find $…

05:24

At noon, ship $A$ is 100 $\mathrm{km}$ west of ship $\mathrm{B}$ . Ship $\ma…