Suppose we imagine the Sun to be about the size of a grapefruit. How big an area would the orbits of the nine planets of the Solar System cover?

size at this scale of the solar system is 10⁸ m²

Explanation:

For this exercise we can use a direct proportions rule or rule of three.

If the radius of the Sun is 7 10⁸ m is equal to the radius of a grapefruit is on average about 5 cm, the radius of the orbit of the plant is x

Mercury

r1 = 5.8 10¹⁰m

x = r1 / r_Sum 5

x = 5.8 10¹⁰/7 10⁸

x = 82 m

We repeat the same formula with all the radii of the orbit, the results in the table

Numb name r_orbit (m) x (m) A (m2)

0 Sun 7 10⁸ 1 3.14

1 mercury 5.8 10¹⁰ 8.2 10¹ 2.0 10⁴

2 venus 1 10¹¹ 1.4 10² 6.2 10⁴

3 Earth 1.5 10¹¹ 2.1 10² 1.4 10⁵

4 Mars 2.3 10¹¹ 3.2 10² 3.2 10⁵

5 Jupiter 7.8 10¹¹ 1.1 10³ 3.8 10⁶

6 Saturn 1.4 10¹² 2 10³ 1.3 10⁷

7 Uranus 2.9 10¹² 4.1 10³ 5.3 10⁷

8 Neptune 4.5 10¹² 6.4 10³ 1.3 10⁸

9 Pluto 5.9 10¹² 8.4 10³ 2.2 10⁸

The area of a circle is

A = π R²

Mercury

A = π 80²

A = 2.0 14 m²

The other values are in the table

The size at this scale of the solar system is 10⁸ m²