Question

The homicide rate decreases at a rate of 8% per year in a city that had 1000 homicides in the most recent year. At this rate, in how many years will the number of homicides reach 500 in a year? Complete parts (a) and (b) below.Give an approximate answer. years (Round to two decimal places as needed.)Give an exact answer. years   enter your response here years (Round to two decimal places as needed.)

Answer 1

ANSWER

$\begin{gathered} 8.3129years \\ \end{gathered}$

EXPLANATION

Given;

Initial Quantity (Ao) = 1000

Rate (r) is 8

The Final Quantity (A(t)) = 500

$r=8=-\frac{8}{100}=-0.08$

Using the rate formula;

$\begin{gathered} A\left(t\right)=A_0(1+r)^t \\ \end{gathered}$

Substituting the values;

$\begin{gathered} A(t)=A_{0}(1+r)^{t} \\ 500=1000(1-0.08)^t \\ 500=1000(0.92)^t \\ \frac{500}{1000}=0.92^t \\ 0.5=0.92^t \end{gathered}$

Using logarithm to solve for t;

$\begin{gathered} t=\frac{log0.5}{log0.92} \\ =\frac{-0.3010}{-0.036} \\ =8.3129 \\ \approx8.31(2d.p) \end{gathered}$

Exact answer is 8.31295.