Problem

The general solution of the differential equation…

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Problem 22 Easy Difficulty

In a culture, the bacteria count is $1,00,000$. The number is increased by $10 \%$ in 2 hours. In how many hours will the count reach $2,00,000$, if the rate of growth of bacteria is proportional to the number present?

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Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 9

Differential Equations

Section 4

Formation of a Differential Equation whose General Solution is given

Discussion

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Video Transcript

but the initial population and we're trying to find the double time. So, um, we are told that at 10 hours the bacteria was at 5000 and then at 12 hours were told that the bacteria was at 6000 and that it's proportional. So we're gonna set up in proportion. We're gonna dio 5000 over 6000 is equal tissue population times E to the 10-K powers. So I was 10 for that top part, and then the initial population times E to the 12 k for 12 hours at the 6000. So the 5000 over six thousands going to reduce to five over X. The initial populations are gonna cancel out for now and weaken Divide eat of the 10-K over each of the 12 K by subtracting the exponents. So 10 minus 12 is negative two. So we're gonna have e to the negative too. Okay, so we're gonna raise both sides to the natural log in order get the e and the natural after cancel, we're gonna divide both sides. I negative to, And what this accomplishes is that gives us our K, which we need in order to find the initial population and the double time. So Natural Lago 56 media close up print senior calculator. Get that answer and divide that by negative too. And you should get a proper points. You're not grounded to do. It's We're gonna take that 0.9 now and we're going to plug it back into one of the equation so you could do 5000 equals p the initial population times E to 10-K or 10 times 100.9 or you do 6000 equals. Finish the population times E to the 12 times point. Oh, night, it doesn't matter. So I'm going to go 5000. So 5000 is equal to the population Trying to find times e raised to the 50000.0, nine times Ted power. And so you're gonna divide both sides by point by eat of the porno nine times 10 and you're gonna want to close that in Brent Theses 0.9 times 10. And if you do that correctly, you should get up initial population of 2032 Now that's not the last thing we need to find. We need to find out the d t the little time. So double time is one divided by 10.9 times the natural log to from art section in the book. And we do want about my 0.9 times natural look of two, you're going approximately 7.7 hours.

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