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 of the base of each cone, \(r = 4\) cm.
- Slant height of each cone, \(l = 6\) cm.
- Height of one cone is \(h\).

Step 1: Use Pythagorean theorem for the cone
In the right-angled triangle formed by the height, radius, and slant height:
\[ l^2 = r^2 + h^2 \] 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 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:
\[ \boxed{4 \sqrt{5} \text{ cm}} \]