Question

a rancher is selling off 7% of her ranch each year. If she owns 2000 square miles, in about how many years will she own only 600 miles

Answer 1

This is an exponential decay function, or it can also be viewed as a geometric sequence, either way, it can be expressed in this circumstance as:

f=ir^t, f=final amount, i=initial amount, r=common ratio or rate, t=times ratio is applied or time.

In this case i=2000 and r=(100-7)/100=0.93 so

f=2000(0.93^t)

We are asked to find when f=600 so:

600=2000(0.93^t)  divide both sides by 2000

0.3=0.93^t, take the natural log of both sides

ln0.3=t*ln0.93  divide both sides by ln0.93

(ln0.3)/(ln0.93)=t

t≈16.59

So in about 16.5 years she will own only 600 square miles.  (round as appropriate for your answer)