Correct option is A
Given:
· Radius of each sphere = 1 unit.
· Five spheres are stacked in a pyramidal structure.
Formula: The total height of the structure = Diameter of the bottom sphere + Radius of the cavity between the lower and upper spheres + Radius of the top sphere.
Solution:
1. The diameter of the bottom sphere is 2 units.
2. The radius of the cavity between the lower and upper spheres is calculated as:
(1 + r)² + (1 + r)² = (1 + 1)²
Simplifying: 2(1 + r)² = 4 (1 + r)² = 2
1 + r = √2
r = √2 - 1.
3. Adding all contributions to the height:
Height = 2 (diameter of the bottom sphere) + 1 (radius of the lower sphere) + (√2 - 1) (radius of the cavity)
Height = 2 + √2.
Final Answer: (a) 2 + √2
