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 triple in size?

How long does it take the population to triple in size?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with solving equations using 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 of T is equal to 200 multiplied by two raised to the quantity of T divided by 24. How long does it take for the fish population to quadruple? We're given four possible choices as our answers. For choice. A we have 60 hours for choice B, we have 36 hours for choice C, we have 72 hours and for choice D, we have 48 hours. Now we're told that the initial population is 200. And so we need that population to quadruple. That means our population at some later time, which is our PFT, it's gonna be equal to four multiplied by 200 which is equal to 800. So that's going to be the population that we're looking for. So now we're just going to use the function that we were given in the problem to find the time. So that function is PFT is equal to 200 multiplied by two raised to the quantity of T divided by 24. So here we'll plug in our 804 RPFT. And so we'll have 800 is equal to 200 multiplying the quantity of two raised to the quantity of T divided by 24. So we need to solve this equation for T. So the first thing I need to do is divide both sides of this equation by 200. And when I do that and simplify, we will get four is equal to two raised to the quantity of T divided by 24. However, we can write four as two squared, so we'll have two squared is equal to two raised to the quantity of T divided by 24. Now, since both of those quantities have the same base of two, that means that their exponents must be equal. So then we have the equation that two is equal to T divided by 24. And we just need to multiply both sides by 24 to isolate T. So doing that, we multiply both sides by 24 and then we'll get 48 is equal to T. So that means that it will take 48 hours in order for the population to quadruple. And if we come back up to our answer choices, it looks like the correct answer choice corresponds to answer choice D. So I hope 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