A plane parallel to the base divides a cone of height 40 cm into two parts. If the volume of the upper smaller cone formed is \(\frac{1}{64}\) of the volume of the original cone, calculate the height of the plane from the base of the cone.
30 cm
For similar cones, the ratio of volumes equals the cube of the ratio of their heights.
Given the upper small cone is 1/64 of the whole: \(\left(\frac{h}{40}\right)^{3}=\frac{1}{64}\).
Taking cube roots: \(\frac{h}{40}=\frac{1}{4}\), so h = 10 cm (the small cone's height measured from the apex).
The plane's height from the base = total height − small cone height = 40 − 10 = 30 cm.
Hence, the answer is 30 cm.
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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 $\pi = \frac{22}{7}$)
