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)

First solve Equation 1 : V₁ + V₂ = 10π.

(4/3)π(r₁)³ + (4/3)π(r₂)³ = 10π

(4/3)π [(r₁)³ + (r₂)³ ] = 10π

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 = 7.18 cubic units.

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