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

Answer 1

The Correct Option is B

Given:
- Radius, \(r = 4\) cm.
- Slant height, \(l = 6\) cm.
- Height of one cone, \(h\).

Step 1: Calculate cone height (h) using Pythagorean theorem
Using the height, radius, and slant height in a right-angled triangle:
\[l^2 = r^2 + h^2\] Substitute values:
\[6^2 = 4^2 + h^2\] \[36 = 16 + h^2\] \[h^2 = 36 - 16 = 20\] \[h = \sqrt{20}\]

Step 2: Simplify \(\sqrt{20}\)
Prime factorization of 20:
\[20 = 4 \times 5\] \[h = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} \text{ cm}\]

Step 3: Calculate total height (H) of the solid
- Two identical cones joined at their bases.
- Total height \(H = h + h = 2h\).
Substitute \(h = 2 \sqrt{5}\):
\[H = 2 \times 2 \sqrt{5} = 4 \sqrt{5} \text{ cm}\]

Final Answer:
\[\boxed{4 \sqrt{5} \text{ cm}}\]