A certain forest covers an area of 2400 km2. Suppose that each year this area decreases by 3% What will the area be after 12 years?

Question

A certain forest covers an area of 2400 km2. Suppose that each year this area decreases by 3% What will the area be after 12 years? Round your answer to the nearest square kilometer.

Answer 1

Before getting into the difficult math, I like setting expectations. This way, we can compare our answers to our expectations and ensure our answers make sense.

Each year, the area is going down by 3%. We'd expect 2400 km² to become slightly smaller after year 1, even smaller after year 2, etc. That means that after 12 years, the area should be noticeably smaller.

Now, let's find out how much smaller it'll be!

Year 0: 2400 km²

Year 1: 3% less than 2400 km² = 2400 - 3%(2400 km²) = 2400 km² - 0.03(2400 km²) = (1 - 0.03)2400 km² = (0.97)2400 km² = 2328 km²

Year 2: 3% less than 3% less than 2400 km² = 3% less than 2328 km² = (0.97)2328 km² = 2258.16 km²

This process is inefficient (and boring!), so let's see if we can create a more general formula using a variable to represent the years passed. Each year that passes leads to a compounding 3% decrease (or a remainder of 97%) each year:

Area = [(0.97)(0.97)(0.97)...](2400 km²) where we'd multiply 0.97 years number of times, or

Area = (0.97)^x(2400 km²) where x is the number of years passed.

When x = 0 and x = 1, Area = 2400 km² and Area = 2258.16 km², respectively.

Finally, plug in x = 12 years passed:

Area = (0.97)^x(2400 km²)

Area = (0.97)¹²(2400 km²) = 1665.22 km² = 1665 km²

This is smaller than I predicted originally, but that shows how powerful exponential functions can be, even with small increases/decreases of 3%!