A cone has height \(h\) and radius \(r\). The cone is melted and recast into a smaller cone whose height is \(\dfrac{h}{2}\) and radius is \(\dfrac{r}{3}\). What fraction of the original volume is unused?
\(\dfrac{17}{18}\)
Volume of original cone:
\(V_1 = \dfrac{1}{3}\pi r^2 h\).
Volume of smaller cone:
\(V_2 = \dfrac{1}{3}\pi \left(\dfrac{r}{3}\right)^2 \left(\dfrac{h}{2}\right) = \dfrac{1}{3}\pi r^2 h \cdot \dfrac{1}{9} \cdot \dfrac{1}{2} = \dfrac{V_1}{18}\).
Unused fraction:
\(\dfrac{V_1 - V_2}{V_1} = 1 - \dfrac{1}{18} = \dfrac{17}{18}\).
Hence the unused fraction is \(\dfrac{17}{18}\).
Similar Questions
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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 = \(22\over7\)]
- 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 $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
