Question

The planets in our solar system can be modeled using spheres. The diameter of Jupiter is 142,984. The volume of Saturn is 59.9% the volume of Jupiter. What is Saturn’s diameter, , in kilometers? Round your answer to the nearest thousandth.

Answer 1

Answer:

120,530 km

Step-by-step explanation:

Vs = Volume of Saturn

Vj = Volume of Jupiter

ds = diameter of Saturn

dj = diameter of Jupiter

Step 1) Vs = 59.9% of Vj

Step 2) Vs = 0.599 * Vj

Now, time to explain why I will do what I will...

Short Lesson: Volumes, areas, and lengths have a special relation. Let's say two figures (figure A and figure B) are similar by a scale factor of k. Then:

Side of figure B = k * Side of figure A

Area of figure B = k² * Area of figure A

Volume of figure B = k³ * Volume of figure A

Back to our example. It's given that the ratio between volumes are 0.599. Therefore, the ratio between the lengths must be ∛0.599. (e.g. if the scale factor of sides are 4, then the scale factor of volumes would be 4³ = 64. Reverse operation tells us if we know the scale of the volumes, we should cube root it to find the sides).

Step 3) ds = $\sqrt[3]{0.599}$ * 142,984

Step 4) ds = 0.842963831 * 142,984

Step 5) ds = 120530 km

Answer 2

Volume of saturn = 59.9% of (volume of Jupiter)
volume of saturn = 0.599*(volume of Jupiter)
volume of saturn = 0.599*(142,984)
volume of saturn = 85,647.416

Use this to plug into the volume of a sphere formula. Solve for r.

V = (4/3)*pi*r^3
85,647.416 = (4/3)*pi*r^3
3*85,647.416 = 3*(4/3)*pi*r^3
256,942.248 = 4*pi*r^3
4*pi*r^3 = 256,942.248
r^3 = (256,942.248)/(4pi)
r^3 = 20,446.8144291719
(r^3)^(1/3) = (20,446.8144291719)^(1/3)
r = 27.3448294160976

The radius is 27.3448294160976 so the diamter is double that at 2*27.3448294160976 = 54.6896588321954 kilometers