Help with this question please!!!

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

S = 150e^-rt where r = annual rate of decrease, t = years with t=0 having 150 Sharks, e = about 2.718281828

(400/3)/150 = e^-2r = 8/9

ln(8/9) = -2r

r = -.5ln(8/9) =about .0589 = about 5.89%

in 2 years 8(150)/9 = 8(50)/3 = 400/3 = 133 1/3= about 133 sharks

in 4 years 8(400)/27 = 3200/27 = about 119 sharks

in 6 years 8(3200)/9(27) =25600/243= about about 105 sharks

in 8 years 8(25600)/243(9) = about 94 sharks

in 10 years 8(8)(25600/243(9)(9) = about 83 sharks

in 12 years 8^3(25600)/243(9^3) = about 74 sharks

in 14 years 8^4(25600)/243(9^4) = about 66 sharks

in 16 years 8^5(25600)/243(9^5) = about 58 sharks

in 18 years 8^6(25600)/243(9^6) = about 52 sharks

in 20 years 8^7(25600)/243(9^7) = about 46 sharks

in 24 years 8^9(25600)/243(9^9) = about 36 sharks

time to reach 35 sharks is a little longer than 24 years

35 = 150e^-.0589t

ln(35/150) = -.0589t

t = ln(7/30)/-.0589 = about -24.70777 years

= about 24 years and 8.49 months

= about 24 years and 8 months when the colony has 35 sharks

24 years means 150(8/9)^12 Sharks

=