Question

At the beginning of an environmental study, a forest covered an area of 1500 km2 . Since then, this area has decreased by 9.8% each year.Let t be the number of years since the start of the study. Let y be the area that the forest covers in km2.

Write an exponential function showing the relationship between y and t.

Answer 1

`A(t)=(0.902)^t \cdot 1500`
`[km^2]`

Explanation:

In this problem, the initial area of the forest at time t = 0 is

`A_0 = 1500 km^2`

After every year, the area of the forest decreases by 9.8%: this means that the area of the forest every year is (100%-9.8%=90.2%) of the area of the previous year.

So for instance, after 1 year, the area is

`A_1 = A_0 \cdot (90.2)/(100)=0.902 A_0`

After 2 years,

`A_2=0.902 A_1 = 0.902(0.902A_0)=(0.902)^2 A_0`

And so on. So, after t years, the area of the forest will be

`A(t)=(0.902)^t A_0`

And by substituting the value of A0, we can find an explicit expression:

`A(t)=(0.902)^t \cdot 1500`
`[km^2]`