Sequences and series question:

Question

A landlocked lake has been selected to be stocked in the year 2020 with trout and to be restocked each year thereafter with trout. Each year the fish population declines due to harvesting as well as natural causes.

Use a graphing utility to find the number of trout as time passes infinitely. Explain your result. Does the population remain constant as time approaches infinity?

Answer 1

Hello, Jen

,

I agree with Mark M. that some information seems to be missing, but I'll demonstrate the approach, and inevitable conclusion.

First, let's look at the formula for calculating the growth, or decline in this case. I'll make the following assumptions:

Set the initial fish population to S and the number of years to x. The fish population, F, declines every year, but we aren't given a value for the magnitude of that decline. We'll assume a percentage decline for this problem. Assign it the symbol "G," which means "growth," but in this case the value will be less than 100%, since the fish population declines every year, despite the restocking effort.

The formula we can use to plot the fish population, F, as a function of years, x, when the growth of that population is G. The formula for compounded growth when we take it year-by-year is:

F(x) = S*(1+G)^x

[The formula is different if the "compounding period is different than 1 year].

I'll assume the following values, so that we can plot this function:

S = 1000 fish at the start, a x = 0

I'll assume the fish decline at a rate of 5% per year. The growth rate, G, is therefore (100-5 = 95%), or 0.95.

G = 0.95

Now we can plot the result of F(x) = 1000*(0.95)^x

My graph won't fit in this box. Go to Desmos and use free graphing software.

It will take slightly over 100 years for the population to decline to almost zero. I calculate that there are 5.92 fish remaining at the end of the 100th year. I'll round that to a whole number after I'm done fishing.

You can take what I've done and use the correct input values. But it is inevitable that a constant decline will result in a curve approaching zero as time increases.

Bob