Answer:
a. The function is a decay function
b. The function is f(n) = 33500 [tex](0.975)^{n}[/tex]
c. The population in the year 2025 is 31,800 to the nearest hundred
Step-by-step explanation:
The form exponential growth function is y = a [tex](1+r)^{x}[/tex] , where
The form exponential decay function is y = a [tex](1-r)^{x}[/tex] , where
∵ In 1995, the population of a town was 33,500
∴ a = 33,500
∵ It is decreasing at a rate of 2.5% per decade
→ The decrease means, it is a decay function, then use the 2nd form above
∴ f(n) = a [tex](1-r)^{n}[/tex] , where f(n) is the population in n decades
∴ It is a decay function
a. The function is a decay function
∵ The rate is 2.5% per decade ⇒ 2.5% each 10 years
∴ r = 2.5%
→ Divide it by 100 to change it to decimal
∵ 2.5% = 2.5 ÷ 100 = 0.025
∴ r = 0.025
→ Substitute the values of a and r in the f(n)
∴ f(n) = 33500 [tex](1-0.025)^{n}[/tex]
∴ f(n) = 33500 [tex](0.975)^{n}[/tex]
b. The function is f(n) = 33500 [tex](0.975)^{n}[/tex]
∵ We need to find the population in 2025
→ Find how many decades from 1995 to 2025
∵ There are 20 years from 1995 to 2025
∵ 1 decade = 10 years
∵ n is the number of decads
∴ n = 2 ⇒ (20/10 = 2)
→ Substitute n in the function by 2
∵ f(2) = 33500 [tex](0.975)^{2}[/tex]
∴ f(2) = 31845.9375
→ Round it to the nearest hundred
∴ f(2) = 31800
c. The population in the year 2025 is 31,800 to the nearest hundred
