Question

the brazilian free - tailed bat can travel 99 miles per hour. after sunset, a colony of bats emerges from a cave and spreads out in a circular pattern. how long before these bats cover an area of 80,000 square miles? use π = 3.14. 0.9 hours 1.6 hours 2.6 hours 5.3 hours

Explanation:

Step1: Recall the area formula of a circle

The area of a circle is given by \( A=\pi r^{2} \), where \( A \) is the area and \( r \) is the radius. We know \( A = 80000\) square miles and \( \pi=3.14 \). First, we solve for \( r \).
\[

$$\begin{align*} 80000&=3.14\times r^{2}\\ r^{2}&=\frac{80000}{3.14}\\ r^{2}&\approx25477.71\\ r&\approx\sqrt{25477.71}\\ r&\approx159.62 \text{ miles} \end{align*}$$

\]

Step2: Relate radius to time using speed

The bats travel at a speed of \( v = 99\) miles per hour. The radius of the circular area is the distance the bats travel from the cave. We use the formula \( t=\frac{d}{v} \), where \( d \) is the distance (radius \( r \)) and \( v \) is the speed.
\[
t=\frac{159.62}{99}\approx1.6 \text{ hours}
\]

Answer:

1.6 hours (corresponding to the option: 1.6 hours)