A culture of bacteria has a population of

Question

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$ , which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$ , where $t$ is the number of hours after the first observation.
How long does it take the population to reach $10,000$ ?

How long does it take the population to reach $10,000$ ?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with solving equations that involve exponents. This problem says a certain species of fish in a lake reproduces at a rate that doubles its population every 24 hours. The initial population of the fish is 200. If the population T hours after the initial observation is represented by the function T of T is equal to 200 multiplied by two raised to the quantity of T divided by 24. How long will it take for the fish population to reach 4000? We give four possible choices as our answers. For choice. A about 112.28 hours for choice B about 96.32 hours. For choice C about 103.73 hours and for choice D about 72.71 hours. Now, we're asked to find how long it will take for the fish population to reach 4000. That means our population at some later time, which will be our PFT is gonna be equal to or 1000. So what we're going to do is use the equation that we were given the problem which is PFT is equal to 200 multiplying two raised to the quantity of T divided by 24. And we're gonna actually solve this equation for time T. So the first thing we'll do is substitute in our value for PFT. And so we'll have 4000 is equal to 200 multiplied by two raised to the quantity of T divided by 24. So we need to solve this equation for T. So the first step is to divide both sides by 200 and then simplify. So on the left hand side, we will have 20 that's gonna be equal to you two raised to the quantity of tea divided by 24. Now, in order to solve for teeth, the first, the next step that I'm gonna take is to take the natural log of both sides. And we're actually gonna use our properties of natural logs to solve for T. So recall that for natural logs, that the natural log of the quantity of B raised to the A power is gonna be equal to a multiplied by the natural log of B. So using that property of natural logs, we're gonna have the natural log of 20 is equal to T divided by 24. And that fraction is multiplied by the natural log of two. So now we just need to do some algebra to isolate T. So we'll divide both sides by the natural log of two. And then we'll also multiply both sides by 24. And doing so will isolate tea. And so what we'll have is 24 multiplied by the quantity of natural log of 20 divided by the natural log of two in quantity is equal to T. And so we can actually plug those values into our calculator and we will get T is equal to 103.73 hours. And here we've rounded to just two decimal places. So that is our time and hours. And so that corresponds to the answer choice. C so I hope that this explanation has been useful and I'll see you in the next video.

My Courses

Chemistry

Biology

Math

Physics

Business

Social Sciences

Programming

Product & Marketing