bacteria-population-if-a-bacteria-population-starts-with-500-bacteria-and-doubles-in-size-every-half

Question

Bacteria population If a bacteria population starts with 500 bacteria and doubles in size every half hour, then the number of bacteria after $t$ hours is $n=f(t)=500 \cdot 4^{t}.$ (See Exercise 1.4 $.30 ) .$
(a) Find the inverse of this function and explain its meaning.
(b) When will the population reach $10,000 ?$

Amrita Bhasin · Accepted Answer

this question asked us to find the inverse of the function and determine when the population will meet. 10,000. Okay, first off for the inverse. We know we can divide by 500. Take the base 10. Divide by law before and we get tea is log of and divided by 500. Divide by log of four part B one Will a population reached 10,000. Simply plug in for 10,000. You&#x27;re gonna need a calculator for this because it&#x27;s really hard to do this on your own. You get 2.1 sex hours.

Heather Zimmers · Answer

okay. The model that were given in equals 100 times two to the Power T over three represents the population of bacteria, the number of bacteria as a function of time. And that function is also called f of tea in this problem, and the first thing we want to do is find the inverse now, Typically, when finding an inverse, we start by switching X and y, and then we solve for y. However, in a real world context, when the variables have meaning like n equals number and T equals time, you don&#x27;t actually want to switch them because you don&#x27;t want end to mean time and t to me number that just won&#x27;t make sense. So instead of that, all we do is isolate the other one. So we went to isolate T. So let&#x27;s divide both sides of the equation by 100 and divided by 100 equals two to the tea over three power. Now we need to use the log. We could use any kind of log we want we could use long base to if we want to. So log, based too of in over 100 equals log based two of two to the tea over three, and that&#x27;s gonna cancel with the two and leave us with tea over three. So to get t by itself, the last thing we need to do is multiply both sides of the equation by three three times Log, based to of an over 100 equals teeth. Now, if we want to, we could call that f inverse of tea. And because we put a T in there, I think I would rather call it f inverse of n to be more consistent with their variables. We could really put any variable in there we wanted, but I want to remember that end stands for the population. So what have we just done? We just found a function that gives us time as a function of population, whereas we started with a function that gave us population as a function of time. So this will be useful if we know the population and we want to find the time. The original one was useful when we knew the time and we wanted to find the population. So let&#x27;s use this to do part B. So here we have our function and the question is, when will the population reach 50,000? So when will end equal 50,000? Find the time so we can substitute 50,000 in for end and self F. Inverse of 50,000 equals three times log based. Two of 50,000 divided by 100. Let&#x27;s go ahead and put that into calculator, and if your calculator is able to do long based, too, that&#x27;s great. If it&#x27;s not able to do long based to you can do three times natural log of 50,000 over 100 divided by natural log of to, and you&#x27;ll get the same answer, and that gives us approximately 26.897 hours until the population hits that number.

Amy Jiang · Answer

Okay, so we&#x27;re asked if I&#x27;d and prime of Steve before we did that. Let&#x27;s multiply out Are actually yeah, that&#x27;s most of what out our power. So we have t minus 10 times T minus son. That&#x27;s equal to t squared minus 20 t and then plus 100. And that&#x27;s all Times three team. So that&#x27;s three t two power off three minus 60 t to depart to plus 300 t and in one plus 40. Okay, so then, in prime of tea within this usar parral, that&#x27;s 922 to minus 1 20 T and plus 300. Okay. And now we want to find the rate at which our population of bacteria changes in eight hours and 11 hours. So that&#x27;s n prime at eight. And and prime acts 11. If you plug this into account later, he gets negative 84 as well as 69

Ankit Gupta · Answer

in this question were given with the mortal 80 equals 300 multiplied by areas to the part. 0.1733 d In the a part, it says we have to find the initial number of bacteria. For that we find a off 00 means we substitute the value off key as zero. So we substitute p a zero on we get dancer s 303 100 initial number of bacteria&#x27;s were there in the be part, it says What is the population after 10 minutes? So it means we find a off then by substituting the value off key as 10 so 300 meters to the par 0.1733 into multiplied by 10. When we calculate this, we get 356 0.76 which is almost equal toe 3 57 bacteria after rounding off in the sea part, it says. How long will it take to double the population? So initial was 300 double will be 600 equals 300 multiplied by ears to the par 0.1733 be on when we divide both sides by 300 we get two equals. It is to the part. 0.1733 d Applying the natural log we get Ln two equals 0.1733 D on finally a tease equal to a land to buy 0.1733 after calculating this figure is equal toe 40 minutes.

Ankit Gupta · Answer

in this question we are given with the model 80 equals 500 multiplied by a raise to the part 0.27 70. In the a part, it says we have to find the initial number off bacteria. So initial number of bacteria we can find out by putting equal to zero. So zero is equal toe 500 multiplied by it is to the par 0.277 to 0. This times becomes zero. So answer comes out to be 500. So 500 initial bacteria over there in the be part, it says we have to find the population after 10 minutes. So we put it, uh, we have to find a 10 equals 500 multiplied by eaters to the par 0.277 multiplied by 10 When we calculate this weekend 659.583 which is almost equal to 6 60 bacteria off. Good rounding off in the sea part it says. How long will it take for the population to double? It means we have to double initial most 500. So double is 1000. So 1000 equals 500. It is to the par 0.27 70. Divide both sides by 500. We get two is equal toe eaters to the par 0.27 70 I&#x27;m applying the natural longer term bigot Ln two is equal to 0.27 70 on. Finally, he&#x27;s equal to a link to divided by 0.277 Calculated this week, it is equal to 25 minutes.

Matthew Lee · Answer

So we need to find the population the rate at which the population is growing when people too, Uh, for the function you have t equals 500 time one 14 over you Buck t squared. And the question is basically asking us to find the value I key prime of two at the rate at which the population is growing. Want equal to find people? T find the derivative. Um so calculate that 1st 500 not going effective. Derivative. So we came down from stone. That was a Frenchie. This part here, Um, you&#x27;re one of them to come here. Oh, in the power of one point. I think you&#x27;re a plan. And then here I&#x27;m gonna need you. The question Drew question role. We have the denominator people teach great square. The old function that squared and a new waiter first in French is the numerator The old numerator 14 you can hire only come for times the denominator If you bought teeth bared my top function 14. I&#x27;m the derivative the denominator which is two teeth in the power rule. Okay, just simplify this 500 time you&#x27;re gone. Accord one Option 50 squared over here we get to be the fork of 200 for Or teeth word monitor a teak word equal. 500 time. It makes my 14th works money. Four teeth squared. Quest 200 over. Deportee acquired squared and five p prime of to talking to into the function. So we&#x27;re gonna have 500 time a negative four two words. Uh, the point. 200 over 15 airports. What up square? Um, before this little bit 500 it&#x27;s gonna make four. And in time, I could afford my 16. Come on. 16 200 one down here for constricting four square, um, and props from Syracuse calculator. And we&#x27;ll get an approximate great, uh, on approximation of Well, I actually finished more defined. Finished the book. Find 500. There&#x27;s gonna be one in 82 um, over 54 squared. Okay, but we have a calculator to approximate this. 500 times. 22. You don&#x27;t get approximately, um 31.21 So, but it needed that. People to population is growing at roughly 31 point to one bacteria per hour.

Problem

When a camera flash goes off, the batteries immed…

Problem 57 Hard Difficulty

Answer

(a) $t=\frac{\log (n / 500)}{\log 4}$
(b) $t \approx 2.16$ hours

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(a) $t=\frac{\log (n / 500)}{\log 4}$(b) $t \approx 2.16$ hours

Biocalculus Calculus for the Life Sciences

Catherine R.

Caleb E.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Catherine R.

Caleb E.

Kristen K.

Michael J.

(a) $t=\frac{\log (n / 500)}{\log 4}$
(b) $t \approx 2.16$ hours

James Stewart

Biocalculus Calculus for the Life Sciences