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 formed by joining the two cones.

Step 1: Find the height of one cone using Pythagoras theorem
For each cone, height \(h\), radius \(r\), and slant height \(l\) 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 the values:
\[ h = \sqrt{6^2 - 4^2} = \sqrt{36 - 16} = \sqrt{20} = 2\sqrt{5} \, \text{cm} \]

Step 2: Calculate the total height of the solid
Since two identical cones are joined at their bases,
Total height \(H = h + h = 2h = 2 \times 2\sqrt{5} = 4\sqrt{5} \, \text{cm}\)

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