A herd of 40 elks is introduced into state game lands. The population of the herd is expected to follow the model.
P(t)=40(a+2t)divided by 1+0.2t
a. determinated the carrying capacity of the herd.
b. what will the population be after 7 years.
Well, when t=inf
P=40 (a/t+2)/(1/t+.2)=80/.2=400
To determine the carrying capacity of the herd, we need to find the value of "a" in the given population model equation: P(t)=40(a+2t)/(1+0.2t)
a. To find the carrying capacity, we need to determine the value of "a" such that the population approaches a stable number as time goes to infinity. At carrying capacity, the population growth rate becomes zero. So, when t approaches infinity, P(t) also approaches a constant value.
As t approaches infinity, the denominator (1+0.2t) approaches infinity, which makes the entire fraction approach zero. Therefore, the expression (a+2t) must approach zero as well.
This means that a = -2t, as t approaches infinity. Therefore, the carrying capacity of the herd is 0.
b. To find the population after 7 years, substitute t = 7 into the population model equation:
P(7) = 40(a + 2(7))/(1+0.2(7))
P(7) = 40(a + 14)/(1+1.4)
P(7) = 40(a + 14)/2.4
To find the population after 7 years, we need to know the value of "a." Unfortunately, the information provided doesn't give us the direct value of "a."
To determine the carrying capacity of the herd, we need to find the limit of the population as time (t) approaches infinity.
a. To find the carrying capacity, we take the limit of P(t) as t approaches infinity.
So, let's calculate the limit of P(t) as t approaches infinity. The equation for P(t) is:
P(t) = 40(a + 2t) / (1 + 0.2t)
Taking the limit as t approaches infinity, we can ignore the terms with the lowest powers of t:
lim(t->∞) P(t) = lim(t->∞) 40(2t) / (0.2t)
= 40(2) / 0.2 (because we can cancel out the t terms)
= 400
Therefore, the carrying capacity of the herd is 400.
b. To find the population after 7 years (t = 7), we can substitute t = 7 into the equation for P(t):
P(7) = 40(a + 2(7)) / (1 + 0.2(7))
We need to know the value of 'a' to calculate the population accurately. Please provide the value of 'a' in order to proceed with calculating the population after 7 years.