Question

Two spheres have surface areas of 100π units2 and 36π units2. If the volume of the larger sphere is π units3, determine the following measures.

scale factor = __
radius of smaller sphere = __
units radius of larger sphere = __
units volume of smaller sphere = __π units3

Answer 1

The formula for the area of a sphere is ...

... A = 4πr²

Solving for r gives

... r = √(A/(4π)) = (1/2)√(A/π)

The formula for the volume of a sphere is ...

... V = (4/3)πr³

Comparing this to the area formula, we see that ...

... V = A·(r/3)

_____

a) Scale factor = √(area ratio) = √(100π/(36π)) = 10/6 = 5/3

... The larger sphere is 5/3 the dimensions of the smaller sphere.

b) Radius of the smaller sphere is

... (1/2)√(36π/π) = 3 . . . units

c) Radius of the larger sphere is

... (1/2)√(100π/π) = 5 . . . units . . . . = (5/3)·3 units

d) Volume of the smaller sphere is

... V = A·r/3 = (36π)·(3/3) = 36π . . . units³

_____

e) Volume of the larger sphere is

... V = A·r/3 = (100π)·(5/3) = 500π/3 . . . units³

Answer 2

Scale factor = 5/3

Radius of smaller sphere = 3 units

Radius of larger sphere = 5 units

Volume of smaller sphere is 36π unit³

Explanation:

Given:

Surface area of larger sphere = 100π

Surface area of smaller sphere = 36π

Volume of larger sphere = π

To find: Scale factor, radius of the spheres, volume of the smaller sphere

Formula used:

Surface area of a sphere = 4πr²

Volume of a sphere = (4/3)πr³

Scale factor = r₁ / r₂

where r₁ and r₂ are radii of the spheres compared.

Solution:

Surface area of larger sphere = 4πr²

100π = 4πr²

r² = 25 => r = 5

∴ Radius of the larger sphere is 5 units.

Surface area of smaller sphere = 4πr²

36π = 4πr²

r² = 9 => r = 3

∴ Radius of the smaller sphere is 3 units.

Scale factor = Radius of larger sphere/ Radius of smaller sphere = 5/3

Volume of smaller sphere = (4/3) πr³ = (4/3)*3³ π = 36π unit³