a) A population of 460 animals that increases at an annual rate of 15%. Write an exponential function to model.
b) Write an exponential function to model. Predict the value of the function after 5 years(nearest whole number). A population of 430 animals that decreases at annual rate of 12%.
c) You decide to buy a boat that costs $ 8700. The normal depreciation for such a boat is 20% per year. What is the value of the boat after 4 years?
a) To model the population of 460 animals that increases at an annual rate of 15%, we can use the general form of an exponential function:
P(t) = P0 * (1 + r)^t
where P(t) is the population at time t, P0 is the initial population, r is the annual growth rate (expressed as a decimal), and t is the number of years.
In this case, the initial population is 460, and the annual growth rate is 15% or 0.15. Therefore, the exponential function to model this population is:
P(t) = 460 * (1 + 0.15)^t
b) To model the population of 430 animals that decreases at an annual rate of 12%, we can modify the previous exponential function by subtracting the rate from 1:
P(t) = P0 * (1 - r)^t
where P(t) is the population at time t, P0 is the initial population, r is the annual decrease rate (expressed as a decimal), and t is the number of years.
In this case, the initial population is 430, and the annual decrease rate is 12% or 0.12. Therefore, the exponential function to model this population is:
P(t) = 430 * (1 - 0.12)^t
To predict the value of the function after 5 years, we substitute t = 5 into the equation:
P(5) = 430 * (1 - 0.12)^5
Simplifying the equation gives the value of the function after 5 years.
c) The value of the boat after 4 years can be calculated by applying the depreciation rate of 20% each year.
To model the depreciation, we can use the general form of an exponential function:
V(t) = V0 * (1 - r)^t
where V(t) is the value of the boat at time t, V0 is the initial value of the boat, r is the annual depreciation rate (expressed as a decimal), and t is the number of years.
In this case, the initial value of the boat is $8700, and the annual depreciation rate is 20% or 0.20. Therefore, the exponential function to model the value of the boat after 4 years is:
V(t) = 8700 * (1 - 0.20)^4
Evaluating the function gives the value of the boat after 4 years.