Question

A city that had 40,000 trees started losing them at a rate of 10 percent per year because of urbanization.

In approximately__ years, the number of trees in the city reduced to a quarter of the original amount. Hint: Model the situation as P = P0(1 − r)t.

Answer 1

Original Number of trees = P₀ = 40,000
Rate of decrease = r = 10% = 0.10

We want to find when the number of trees will be one quarter of the original number. One quarter can be expressed as 25%, so we want to find when the number of trees is 25% of P₀ .

Using the given equation, we can write:


`P= P_(o)(1-r)^(t) \\ \\ 0.25 P_(o)=P_(o)(1-0.1)^(t) \\ \\ 0.25= (0.9)^(t) \\ \\ log(0.25)=log((0.9)^(t)) \\ \\ log(0.25)=t*log(0.9) \\ \\ t= (log(0.25))/(log(0.9)) \\ \\ t=13.1`

Thus, in approximately 13 years, the number of trees in the city will be reduced to a quarter of the original amount.