Four identical incompressible spheres of radius 1 unit are stacked in a pyramidal form as shown in the figure. All the spheres touch each other. The height of the structure is

Correct option is A

Solution:

$$\begin{aligned}&\text{Let radius } r = 1 \Rightarrow \text{Distance between centers} = 2 \\[5pt]&\text{Three bottom spheres form an equilateral triangle of side } 2 \\[5pt]&\text{The 4th sphere is on top, forming a regular tetrahedron of side 2} \\[5pt]&\text{Height of tetrahedron } h = \frac{\sqrt{6}}{3} \cdot 2 = \frac{2\sqrt{6}}{3} \\[5pt]&\text{Total height of the structure } = 1 + \frac{2\sqrt{6}}{3} + 1 = 2 + \frac{2\sqrt{6}}{3} \\[5pt]\end{aligned}$$

$$\boxed{2 + 2\sqrt{\frac{2}{3}} = 2 + \frac{2\sqrt{6}}{3}}$$​​

​Final Answer: Option A