Please help me answer thiss!!

Question

A colony of sharks is losing one-ninth of its population every two years. The colony currently has 150 sharks.

a.) Find the annual rate considering the fact that P=8/9P0 after 2 years by solving 8/9P0=P0e-2r for r

b.) Find an exponential function of the form P=P0e-rt to model the colony’s growth using your rate from part (a) and the fact that the current population is 150.

c.) How many sharks will be in the colony four years from now?

d.) In how many years (to the nearest month) will the colony have 35 sharks?

Answer 1

shark population decreases by 1/9 every 2 years

(8/9)Po = Po(e^2r)

8/9 = e^2r

ln(8/9) = 2r

r = ln(8/9)2 = about -0.0589 = -5.89% annual decrease

S= 150e^-.0589t = shark population in year t

in 2 years S=8(150)/9

in 4 years S= 7(150)/9 = 1050/9 = 116 2/3 = about 117 sharks left

or

S =150e^-.0589(4) = 150e^-.2356= 118.5 = about 119

117 is more accurate, as r=.0589 was rounded off with r slightly less than .0589

35 = 150e^-.0589t

35/150 = 7/30 = e^-.0589t

ln(7/3) = -.0589t

t = ln(7/3)/-.0589 = 24.708 years

= 24 years and .708(12) months

= 24 years and 8.49 months

= 24 years, 8 months