A solid right circular cone is 7 cm high, and the radius of its base is 22.2 cm. It is melted and recast into a right circular cone with radius of its base 3.7 cm. Then the height (in cm) of the cone is ______.
252
Melting and recasting keeps the volume unchanged. The volume of a cone is \(V=\frac{1}{3}\pi r^2 h\), so \(r_1^{2}h_1=r_2^{2}h_2\).
Substitute the known values: \((22.2)^2\times 7=(3.7)^2\times h_2\).
Note that \(22.2=6\times 3.7\), so \((22.2)^2=36\times(3.7)^2\).
The equation becomes \(36\times(3.7)^2\times 7=(3.7)^2\times h_2\); cancelling \((3.7)^2\) gives \(h_2=36\times 7\).
Therefore \(h_2=252\) cm.
Hence, the answer is 252.
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
