The park currently contains 20 wolves and the park rangers are introducing additional wolves at the rate mt(12−t) wolves per month, 0≤t≤12

for some constant m>0 to be determined. However, the park rangers are concerned about damaging the current rabbit population. It is estimated that the rate of growth of the rabbit population in the park is given by
r(t)=56+10cos(πt6)−W(t) rabbits per month, 0≤t≤12
where W(t) is the number of wolves in the park at time t.
(a) Write an expression for the function W(t) for 0≤t≤12.
(b) What constant m should the park rangers choose so that the rabbit population is the same at the end of the year as in the beginning of the year?
(c) With the same constant m of part b), what is the average rate at which wolves will be introduced to the park?

(a) To write an expression for the function W(t), we need to understand the given information. The park currently contains 20 wolves, and additional wolves are being introduced at a rate of mt(12−t) wolves per month. We need to determine the expression for W(t).

We know that initially, at t = 0, there are 20 wolves in the park. We can represent this as W(0) = 20.

The rate at which wolves are being introduced is mt(12−t). Here, t represents time, where 0 ≤ t ≤ 12.

Therefore, the expression for the function W(t) for 0 ≤ t ≤ 12 can be written as:

W(t) = 20 + ∫[0,t] (mt(12−t)) dt

(b) To ensure that the rabbit population remains the same at the end of the year as in the beginning of the year, we need to find the constant m.

The rabbit population is given by the function r(t) = 56 + 10cos(πt/6) − W(t).

At the beginning of the year, t = 0, the rabbit population is r(0). At the end of the year, t = 12, the rabbit population is r(12). Since we want these populations to be the same, we can set r(0) equal to r(12) and solve for m.

r(0) = r(12)

56 + 10cos(0) − W(0) = 56 + 10cos(π) − W(12)

Simplifying, we get:

56 - W(0) = 56 - W(12)

Since we know that W(0) = 20, we can substitute and solve for W(12):

56 - 20 = 56 - W(12)

36 = W(12)

Now, we can substitute this value of W(12) into the expression for W(t) from part (a), and solve for m:

W(t) = 20 + ∫[0,t] (mt(12−t)) dt

W(12) = 20 + ∫[0,12] (m(12−t)) dt

36 = 20 + ∫[0,12] (m(12−t)) dt

16 = ∫[0,12] (m(12−t)) dt

16 = m ∫[0,12] (12−t) dt

16 = m [12t - (t^2/2)] |[0,12]

16 = m [(12*12 - (12^2/2)) - (0 - (0^2/2))]

16 = m [144 - 72]

16 = m * 72

m = 16/72

Simplifying, we find:

m = 2/9

Therefore, the constant m should be 2/9 for the rabbit population to remain the same at the end of the year as in the beginning of the year.

(c) With the same constant m as in part (b), the average rate at which wolves will be introduced to the park can be calculated by finding the average value of the function mt(12−t) over the interval 0 ≤ t ≤ 12.

The average value can be calculated using the integral of the function divided by the length of the interval.

Average rate = (1/(12-0)) * ∫[0,12] (mt(12−t)) dt

Average rate = (1/12) * ∫[0,12] (2/9 * t(12−t)) dt

Average rate = (2/108) * ∫[0,12] (t(12−t)) dt

Average rate = (1/54) * ∫[0,12] (t(12−t)) dt

Evaluating the integral, we get:

Average rate = (1/54) * [(t^3/3) * (12 - t^2/2)] |[0,12]

Average rate = (1/54) * [(12^3/3) * (12 - (12^2/2)) - (0^3/3) * (12 - (0^2/2))]

Average rate = (1/54) * [(1728/3) * (12 - 72) - 0]

Average rate = (1/54) * [1728 * (-60)]

Average rate = (1/54) * (-103680)

Average rate = -1920

Therefore, with the constant m = 2/9, the average rate at which wolves will be introduced to the park is -1920 wolves per month.