A solid hemisphere of radius R is melted and recast into n smaller hemispheres of radius r. Find the value of n.
\(\left(\dfrac{R}{r}\right)^3\)
When a solid is melted and recast, the total volume is conserved.
Volume of a hemisphere of radius \(a\) is \(\dfrac{2}{3}\pi a^3\).
Conservation of volume:
\(\dfrac{2}{3}\pi R^3 = n \times \dfrac{2}{3}\pi r^3\)
Cancel the common factor \(\dfrac{2}{3}\pi\):
\(R^3 = n\, r^3 \;\Longrightarrow\; n = \dfrac{R^3}{r^3} = \left(\dfrac{R}{r}\right)^3\)
Hence \(n = (R/r)^3\) — option (1).
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Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$
