The game commission introduces 100 deer into newly acquired state game lands. the population N of the herd is modeled by N=(20(5+3t))/1+0.04t t>/=0
where t is the time in years
a. find the population when t=5, t=10, and t=25
b. what is the limiting size of the herd as time increases?
you must mean
N = 20(5+3t)/(1+.04t)
plug in t = 5
N = 20(5+15)/(1 + .04(5))
= 400/1.02
= 2000
Do the others likewise.
as t ---> large
n = (100 + 60(large))/(1 + .04 large)
= appr 60large/.04large
= appr 60/.04 ---> 1500
the limiting size = 1500
e.g. let t = 100,000
N = 20(300005)/(1 + 4000) = 1500.025
a. Sure, let me plug in the values for t and calculate the population:
For t = 5:
N = (20(5 + 3(5))) / (1 + 0.04(5))
N = (20(5 + 15)) / (1 + 0.2)
N = (20(20)) / (1.2)
N ≈ 333.33 (Deer don't do well with fractions, so let's call it 333)
For t = 10:
N = (20(5 + 3(10))) / (1 + 0.04(10))
N = (20(5 + 30)) / (1 + 0.4)
N = (20(35)) / (1.4)
N ≈ 500 (Deer like round numbers)
For t = 25:
N = (20(5 + 3(25))) / (1 + 0.04(25))
N = (20(5 + 75)) / (1 + 1)
N = (20(80)) / (2)
N = 800 (Deer are really good at math)
b. Ah, the limiting size of the herd as time increases. Well, as time goes on, the denominator of the equation, (1 + 0.04t), gets bigger and bigger. That means the population approaches infinity! Or at least a really large number. So, theoretically, the herd size keeps growing forever. Can you imagine a world full of endless deer? It would be, dare I say, quite "deer"ful.
To find the population at different time intervals and the limiting size of the herd, we will substitute the values of t into the equation N = (20(5 + 3t))/(1 + 0.04t).
a. Population at t = 5 years:
N = (20(5 + 3(5)))/(1 + 0.04(5))
N = (20(5 + 15))/(1 + 0.2)
N = (20(20))/(1.2)
N = 400/1.2
N ≈ 333.33 (rounded to 2 decimal places)
Population at t = 10 years:
N = (20(5 + 3(10)))/(1 + 0.04(10))
N = (20(5 + 30))/(1 + 0.4)
N = (20(35))/(1.4)
N = 700/1.4
N = 500
Population at t = 25 years:
N = (20(5 + 3(25)))/(1 + 0.04(25))
N = (20(5 + 75))/(1 + 1)
N = (20(80))/(2)
N = 1600/2
N = 800
So, the population at t = 5 years is approximately 333.33, at t = 10 years is 500, and at t = 25 years is 800.
b. To determine the limiting size of the herd as time increases, we evaluate the value of N as t approaches infinity. When t becomes very large, the term involving t in the denominator becomes negligible compared to the other terms. Therefore, we can approximate the limiting size by looking at the behavior of the expression without the t term in the denominator:
lim(N) as t approaches infinity ≈ lim((20(5 + 3t))/(1)) as t approaches infinity
lim(N) as t approaches infinity ≈ 20(5 + 3t)
lim(N) as t approaches infinity ≈ 20 * 3t (ignoring the constant term and multiplying)
lim(N) as t approaches infinity ≈ 60t
Hence, the limiting size of the herd as time increases is given by the expression 60t. This means that the herd size will continue to increase indefinitely as time goes on.