Question

1. Jonathan is building a chicken farm in 2023. The initial population of his farm is 2850 chickens. The population of his chicken farm grows at a rate of 3% annually.

(a) Write an exponential equation that can be used model the population of the farm t years after 2023.
(b) Using this equation, estimate the population of the chicken farm in 2045. Please round to the nearest chicken (no partial chickens, please!)

Answer 1

(a) Let P(t) be the population of the chicken farm t years after 2023. Then the population grows at a rate of 3% annually, which means that the population is multiplied by a factor of 1.03 each year. Therefore, the exponential equation that models the population is:

P(t) = 2850(1.03)^t

(b) To estimate the population of the chicken farm in 2045, we need to find P(22), where t is the number of years from 2023 to 2045 (which is 22 years). We can use the equation from part (a) to calculate this:

P(22) = 2850(1.03)^22

≈ 4849.7

Therefore, the estimated population of the chicken farm in 2045 is about 4849 chickens. Rounded to the nearest whole chicken, this is 4850 chickens.

Answer 2

Explanation:

(a) The exponential equation that can be used to model the population of the chicken farm t years after 2023 is:

P(t) = P₀(1 + r)ᵗ

where:

P(t) = the population of the farm t years after 2023

P₀ = the initial population of the farm (2850 chickens)

r = the annual growth rate (3% or 0.03, expressed as a decimal)

ᵗ = the time in years

Therefore, the equation is:

P(t) = 2850(1 + 0.03)ᵗ

(b) To estimate the population of the chicken farm in 2045, we need to substitute t = 22 into the equation and solve for P(22):

P(22) = 2850(1 + 0.03)²²

P(22) ≈ 5242 chickens

Therefore, the estimated population of the chicken farm in 2045 is 5242 chickens (rounded to the nearest chicken)