Question

Suppose there are approximately 4.0 × 106 asteroids in the asteroid belt that have a diameter larger than 0.3 km. If the asteroid belt stretches from 2.1 AU to 3.9 AU, how much space is there for each asteroid? Follow the steps below. (There are 1.49 × 108 km in 1 AU.)

HINT(The asteroid belt is a very desolate place. The area of a circle with a radius of R is area= 3.14 x R^2

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

1)
R1 = 2.1 AU = 2.1 * 1.49*10^8 km = 3.129*10^8 km
R2 = 3.9 AU = 3.9 * 1.49*10^8 km = 5.811*10^8 km
Area of belt = 3.14* (R2^2 - R1^2)
= 3.14 *((5.811*10^8 )^2 - (3.129*10^8 )^2 )
= 7.53*10^17 Km^2

2)
Number of asteroids = 4*10^6
Area for each =(7.53*10^17) /(4*10^6)
= 1.88*10^11 km^2

3)
Number = area of each asteroid / area of Australia
= (1.88*10^11 km^2) / (7.74*10^6 km^2)
= 2.4*10^4