Question

There is a cube Alpha whose side is equal to the diagonal of the cube Beta of side 2. What is the ratio of the volume of the largest sphere Delta, inside the cube Gamma of side equal to the diagonal of cube Alpha and volume of cube Beta?

• A. 2π : 9
• B. 8π : 9
• C. 9π : 8
• D. 9π : 2

Answer 1

The correct option is D

9π : 2

Option (d)
Side length of Beta cube = 2 cm
Volume of Beta cube = (2)${}^3$ = 8 cm${}^3$ .........-(1)
Diagonal length of Beta cube = side length of Alpha cube = 2$\sqrt{3}$ cm
Side length of Gamma cube = diagonal length of Alpha cube = (2 $\sqrt{3}$)$\sqrt{3}$ = 6 cm.
Diameter of sphere inside Gamma cube = Length of edge = 6 cm.
Radius of sphere =$\frac{6}{2}$ = 3 cm
Volume of sphere S
= $\left(\frac{4}{3}\right)\times \pi \times (3{)}^3$ = 36$\pi$.........(2)
Required ratio = S : q = 36$\pi$ : 8 = 9$\pi$ : 2