Question

Suppose there are approximately 8.0 × 106 asteroids in the asteroid belt that have a diameter larger than 0.2 km. If the asteroid belt stretches from 2.1 AU to 3.7 AU, how much space is there for each asteroid? Follow the steps below. (There are 1.49 × 108 km in 1 AU.)1) Find the area of the asteroid belt. 2) Calculate how much area there is for each asteroid. 3) Australia has an area of 7.74 × 106 km2. How many areas the size of Australia does each asteroid have?

Answer 1

The area of the asteroid belt is given by:

$A=\pi(R^2-r^2)=\pi[(3.7\times1.49\times10^8km)^2-(2.1\times\times(1.49\times10^8)km)^2]$

$A=6.4724\times10^{17}\: \: km^2$

The available area for each asteroid is:

$A_{each}=\frac{6.4724\times10^{17}\: \: km^2}{8.0\times10^6asteroids}=8.0\times10^{10}\: \: km^2/asteroid$

How many areas the size of Australia does each asteroid have?

$\frac{8.0\times10^{10}\: \: km^2}{7.74\times10^6\: \: km^2}=10452.96$

Each asteroid has 10452.96 areas the size of Australia