What is the ratio V2/V1 of the volumes?

[tenchick19]

I need help with this question: Sphere 1 has surface area A1 and volume V1, and Sphere 2 has surface area A2 and volume V2. If the radius of Sphere 2 is six times the radius of Sphere 1, what is the ration A2/A1 of the areas?

Part two: What is the ratio V2/V1 of the volumes?

 

[TD]

For a sphere with radius R, area (A) and volume (V) are:

$$\begin{array}{l} A = 4\pi R^2 \\ V = \frac{4}{3}\pi R^3 \\ \end{array}$$

With your knowledge of the ratio's of the radii, that should help

 

[quicknote]

I can provide a mathematical explanation for the ratio of volumes V2/V1. The volume of a sphere is calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. Therefore, the ratio of volumes V2/V1 can be calculated as (4/3)πr2^3 / (4/3)πr1^3, where r2 is the radius of Sphere 2 and r1 is the radius of Sphere 1.

Substituting the given information that the radius of Sphere 2 is six times the radius of Sphere 1, we get (4/3)π(6r1)^3 / (4/3)πr1^3. Simplifying this expression, we get (6^3) / 1, which is equal to 216. Therefore, the ratio of volumes V2/V1 is 216:1.

For the second part of the question, the ratio of areas A2/A1 can be calculated using the formula A = 4πr^2, where r is the radius of the sphere. Substituting the given information, we get A2/A1 = (4π(6r1)^2) / (4πr1^2), which simplifies to 36:1. Thus, the ratio of areas A2/A1 is 36:1.

In summary, the ratio of volumes V2/V1 is 216:1 and the ratio of areas A2/A1 is 36:1.