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 3.5 times the radius of sphere 1, what is the ratio of each of the following?
Incomplete.
To solve this problem, we'll use the formulas for the surface area and volume of a sphere.
The formula for the surface area of a sphere is A = 4πr^2, where A is the surface area and r is the radius.
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.
Let's start by finding the ratio of the surface areas of the two spheres.
We're given that the radius of sphere 2 (r2) is 3.5 times the radius of sphere 1 (r1). So we can express this relationship as r2 = 3.5r1.
The surface area of sphere 1 is A1 = 4πr1^2.
The surface area of sphere 2 is A2 = 4πr2^2.
Now, substitute r2 = 3.5r1 into the equation for A2 to get A2 = 4π(3.5r1)^2 = 4π(12.25r1^2) = 48.5πr1^2.
So the ratio of the surface areas is A2/A1 = (48.5πr1^2)/(4πr1^2) = 12.125.
Next, let's find the ratio of the volumes of the two spheres.
The volume of sphere 1 is V1 = (4/3)πr1^3.
The volume of sphere 2 is V2 = (4/3)πr2^3.
Substitute r2 = 3.5r1 into the equation for V2 to get V2 = (4/3)π(3.5r1)^3 = (4/3)π(42.875r1^3) = 57.17πr1^3.
So the ratio of the volumes is V2/V1 = (57.17πr1^3)/((4/3)πr1^3) = 42.8775.
Therefore, the ratio of the surface areas is 12.125 and the ratio of the volumes is 42.8775.