Question

The total volume of two spheres is 10π cubic units. The ratio of the

areas is 4:9. What is the volume of the smaller sphere in cubic units?

Answer 1

The problem requires using the formulas for a sphere's volume and surface area, along with the given total volume and surface area ratio, to deduce the volume of the smaller sphere.

Step-by-step explanation:

To solve this problem, we will use the formulas for the volume and surface area of a sphere and the given ratio to find the volumes of the two spheres.

The volume (V) of a sphere is given by the formula V = (4/3) π r³, and the surface area (A) is given by the formula A = 4 π r². Given the total volume of the two spheres being 10π and the ratio of their surface areas as 4:9, we can set up equations to solve for the radii of the two spheres.

Let's assume the radius of the smaller sphere is r and the radius of the larger sphere is R. From the surface area ratio, we have:

• 4 π r² / 4 π R² = 4/9
• r² / R² = 1/2.25
• r / R = 1/1.5 or r = (2/3) R

Substituting r = (2/3) R into the volume formulas:

• Vsmall = (4/3) π (2/3R)³
• V large = (4/3) π R³
• Vsmall + Vlarge = 10 π

Solving these equations, we find the volumes of the individual spheres and hence the volume of the smaller sphere.

Answer 2

4 r1 ² π : 4 r2² π = 4 : 9
r1² : r2² = 4 : 9
r1 : r2 = 2 : 3
r2 = 3 r1/2 ( r - radius of the smaller sphere )
10 π = 4/3 r³ π + 4/3 ( 3 r/2 ) ³ π / : π/3
30 = 4 r³ + 4 * 27 r³/8
30 = 4 r³ + 27 r³/2 / * 2
60 = 35 r³
r³ = 60 / 35 = 12/7
V ( Smaller ) = 4/3 r³ π = 4/3 * 12 / 7 π = 16/7 π = 2.286 π cubic units