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JH
Numerade Educator

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Problem 67 Hard Difficulty

A fish has 5000 catfish in his pond. The number of catfish by $ 8 \% $ per month and the farmer harvests 300 catfish per month.
(a) Show that the catfish population $ P_n $ after $ n $ months is given recursively by
$ P_n = 1.08 P_{n-1} - 300$
$P_0 = 5000$
(b) How many catfish are in the pond after six months?

Answer

a. $P_{n}=1.08 P_{n-1}-300$
b. 5734

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Video Transcript

so initially we have 5000 catfish in the pond. However, that will increase by 8%. But then we also lose an additional 300 fish per month due to harvest. So here we have P N for part A. So what's the amount? After n months? Well, we take the previous amount from last month, but then we'll also add 8% of that. How and because of harvest were also subtracting off 300. And then if we just simplify this, this is just 1.8 p n minus one minus 300. This verifies the formula in part A for part B. Let's just use this formula repeatedly to find B six p six. Excuse me so I'll need to go to the next page for this one p one That's just 1.8 p zero minus 300 So 5100 p. Two. 1.8 p. One minus 300 52 Oh eight. Let's keep going in this fashion P three 1.8 52 oh eight minus 300 this time for P three 53 to 4 0.64 Let's not worry about the decimals right now. Let's just keep them until we get to the very end. Then we'll figure out what to do with the decimal. Now, I'm just going to carry this decimal over minus 300 so I'm just using the formula over and over again. This time I'm on P four, 5450 6112 two more to go minus 300 and pf five. That's five, 586 66 Oh, 96 And finally P six. And then we have P five here minus 300. And then take this one back to the calculator. And this one is 5733 with some change left over. That's about 0.59 and we should round this up to be set 5734 And the units were catfish. So after six months, we have 5734 catfish after six months, and that's our final answer