Question

Two identical cones are joined as shown in the figure. If radius of base is 4 cm and slant height of the cone is 6 cm, then height of the solid is

• A. 8 cm
• B. \(4\sqrt{5}\) cm
• C. \(2\sqrt{5}\) cm
• D. 12 cm

Hint:

Use the Pythagoras theorem: \(l^2 = r^2 + h^2\) in right-angled triangles of cones.

Answer 1

The Correct Option is B

Given:
Two identical cones joined at their bases.
Radius of base, \(r = 4 \, \text{cm}\)
Slant height of each cone, \(l = 6 \, \text{cm}\)

To find:
Height of the solid.

Step 1: Find cone height using Pythagoras.
For each cone, \(h\), \(r\), and \(l\) form a right triangle:
\[l^2 = r^2 + h^2\Rightarrow h^2 = l^2 - r^2\Rightarrow h = \sqrt{l^2 - r^2}\]
Substitute values:
\[h = \sqrt{6^2 - 4^2} = \sqrt{36 - 16} = \sqrt{20} = 2\sqrt{5} \, \text{cm}\]

Step 2: Calculate total height.
Two identical cones joined at their bases:
Total height \(H = h + h = 2h = 2 \times 2\sqrt{5} = 4\sqrt{5} \, \text{cm}\)

Final Answer:
Height = \(4\sqrt{5}\) cm