Pre-Calcus Question: Determine how many mosquitos / raccoons are in the area

Question

The recent opening of a landfill in the area has caused the raccoon population to flourish, with an adverse effect on the number of purple martins (type of bird). Wildlife specialists believe the population of purple martins p will decrease as the population of raccoons, r, grows. The relationship between these two populations is represented by the equation p(r)= 750 - 3.75r. Mosquitos are the primary diet of purple martins (type of bird). Thus, the mosquito population m is a function of the population of purple martins p and is represented by the equation m (p) = 50,000 -45p

a. if there were 200 raccoons in the area 6 months after the landfill opened, then how many mosquitoes are in the area at that time?

b. if there were r raccoons in the area 1 year after the landfill opened, then how many mosquitoes are in the area at that time?

c. if we don't want more than 36,500 mosquitoes, then what should the raccoon population be?

Answer 1

The easiest way to answer these questions would be to have a function where mosquitoes are a function of raccoons. To get that, we simply combine the functions we have. Since m(p) = 50,000 - 45p and p(r) = 750 - 3.75r we can just plug in "750 - 3.75r" into the first function wherever we see a "p". That makes

m(r) = 50,000 - 45(750 - 3.75r)

m(r) = 50,000 - 33750 + 168.75r

m(r) = 16,200 + 168.75r

So if there are 200 raccoons, there are 16,200 + 168.75(200) or 50,000 mosquitoes

If there are r raccoons, then there are 16,200 + 168.75r mosquitoes

If we want no more than 36,500 mosquitoes, we plug that in and calculate the associated number of raccoons.

36,500 = 16,200 + 168.75r

20,300 = 168.75r

r = 120.3

Of course, we can't have 0.3 of a raccoon so we must have 120 raccoons or less to keep the population of mosquitoes to a max of 36,500

Answer 2

m(p(r)) = m(750 - 3.75r) = 50000 - 45(750 - 3.75r) = 16250 + 168.75r

a. If r = 200, then m = 16250 + 168.75(200) = 50000 mosquitoes

b. If r raccoons are present, then the number of mosquitoes is 16250 + 168.75

c. If m ≤ 36500, then 16250 + 168.75r ≤ 36500

168.75r ≤ 20250

r ≤ 120