Question

A little-known species of insect is approaching extinction, with a population that falls by 10%

every year. There are currently 2,400 insects remaining. How many will there be in 4 years?

Answer 1

• Rate of fall=10%=0.1
• Current=2490
• Time=4yr

The equation is

• y=ab^x
• y=(2400)(1-0.1)^4
• y=2400(0.9)⁴
• y=1574.6

Approximately 1575

Answer 2

approximately 1,575 insects

Explanation:

We can model this as an exponential equation.

General form of an exponential equation:
`y=ab^x`

where:

• a is the initial value)
• b is the base (or growth/decay factor)
• x is the independent variable
• y is the dependent variable

If b > 1 then it is an increasing function

If 0 < b < 1 then it is a decreasing function

If the population falls by 10% each year, then each year the population will be 90% of the previous year (since 100% - 10% = 90%)

Convert 90% into a decimal: 90/100 = 0.9

Therefore, the base of the exponential function is 0.9

Given:

• a = 2400
• b = 0.9
• x = time (in years)
• y = population of insects

`\implies y=2400(0.9)^x`

To find how many insects there will be in 4 years, substitute x = 4 into the equation and solve for y:

`\begin{aligned}x=4 \implies y & =2400(0.9)^4\\& = 1574.64\end{aligned}`

Therefore, there will be approximately 1,575 insects remaining in 4 years.