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Problem 56

Population Growth The rate of growth $d P / d t$ of a population of bacteria is proportional to the square root of $t,$ where $P$ is the population size and $t$ is the time in days $(0 \leq t \leq 10) .$ That is,

$\frac{d P}{d t}=k \sqrt{t}$

The initial size of the population is $500 .$ After 1 day, the population has grown to $600 .$ Estimate the population after 7 days.

$$\begin{array}{l}{f^{\prime \prime}(x)=\sqrt{x}} \\ {f^{\prime}(x)=\frac{3}{4} x^{\frac{4}{3}}} \\ {f(x)=\frac{9}{28} x^{\frac{7}{3}}}\end{array}$$

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## Video Transcript

so. Oh, we have duping me, too. People, too. Room two on the piers Hero. This 500. Well, taking the time from Pune. You too. You too. You should have. Yeah, Okay. Two Tintin choose. Okay, times 2/3. 223 rounds close. Some constancy with as a team appear. Zero. It happens to be 500. Is he quitting? Okay, Time's 2/3 time. Zero 23 hours close. So seems 500. Yeah, So on the seventh day Sorry. We skipped ahead a bit too far. I know that one 600 and that's T K over three times. 12123 hives. Class 500. Taking the 500 from both sides. We have 100. Simple 2/3. Can clients that okay is 300 over to which is 150. Let me your team. It's equal to 150 over three times two just 100 23 hours, US. So that implies that seven. See, once you know, 100 terms sermon to the three hives. Close 500. Put the make sense