Arancher 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

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)

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