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

Question

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

 

 

Answer 1

The ratio of areas is 4:9.

Formula for area of the sphere is 4πr².

Area of first sphere / Area of second sphere = 4πr₁² / 4πr₂² = 4/9.

(r₁ / r₂)² = 4/9

r₁ / r₂ = √(4) / √(9)

r₁ / r₂ = √(2²) / √(3²)

r₁ / r₂ = 2 / 3 = 0.6667 units.

Volume of the sphere = V₁ = 4/3 π (r )³

Volume of first sphere / Volume of second sphere = V₁ / V₂

= [4/3 π (r₁)³ ] / [4/3 π (r₂)³]

= (r₁)³ / (r₂)³

= (r₁ / r₂)³

= (0.6667)³

= 0.2962 cubic units.

Total volume of the spheres = 10π.

Let V₁ = 10 - V₂.

Therefore  V₁ / V₂ = (10 - V₂) / V₂ = 0.2962

10 / V₂ - V₂ / V₂ = 0.2962

10 / V₂ - 1 = 0.2962

10 / V₂ = 0.2962 + 1

10 / V₂ = 1.2962

10 / 1.2962 = V₂

7.7148 = V₂

Therefore volume of smaller sphere is  V₁ = 10 - V₂ = 10 - 7.7148 = 2.2852 cubic unit.

Comments

Total volume of the spheres = 10π.

The equation is V₁ + V₂ = 10π  ⟹ V₁ = 10π - V₂.

V₁ = 16π/7 = [16(22/7)]/7 ≅ 7.18 cubic units.

V₂ = 54π/7 = [54(22/7)]/7 ≅ = 24.24 cubic units.

Answer 2

The formula for the volume and the area are V = (4/3)πr³ and A = 4πr² respectively, where r = radius.

The total volume of two spheres is 10π cubic units and the ratio of the areas is 4 : 9.

Let V₁, A₁ and r₁ be the volume, area and radius of the smaller sphere, and V₂, A₂ and r₂ be the volume, area and radius of the larger sphere.

From the information,

V₁ + V₂ = 10π --------> (1)

A₁ : A₂ = 4 : 9 --------> (2)

To find the radii of two spheres, solve equation 2 : A₁ : A₂ = 4 : 9.

4π(r₁)² : 4π(r₂)² = 4 : 9

(r₁)² / (r₂)² = 2² / 3².

(r₁ / r₂)² = (2 / 3)².

r₁ : r₂ = 2 : 3.

Let r₁ = 2x and r₂ = 3x ⟹ (r₁)³ = 8x³ and (r₂)³ = 27x³.

To find the value of x or x³, substitute values of r₁ = 2x and r₂ = 3x in the equation 1 : V₁ + V₂ = 10π.

(4/3)π(2x)³ + (4/3)π(3x)³ = 10π

(4/3)π(8x³)+ (4/3)π(27x³)= 10π

(4/3)π(x³) [ 8 + 27] = 10π

(4/3)π(x³) [35] = 10π

x³ = 3/14.

(r₁)³ = 8x³  ⟹  (r₁)³ = 8(3/14)  ⟹  (r₁)³ = 12/7.

(r₂)³ = 27x³  ⟹  (r₂)³ = 27(3/14)  ⟹  (r₁)³ = 81/14.

Volume of the smaller sphere V₁ = (4/3)π(r₁)³ = (4/3)π(12/7) = 16π/7= [16(22/7)]/7 ≅ 7.18 cubic units.

Volume of the larger sphere V₂ = (4/3)π(r₂)³ = (4/3)π(81/14) = 54π/7 = [54(22/7)]/7 ≅ = 24.24 cubic units.