wax cylinder of diameter 21cm and height 21 cm is chipped off and shaped to form a cone of maximum - Maths - Surface Areas and Volumes

Question

wax cylinder of diameter 21cm and height 21 cm is chipped off
and shaped to form a cone of maximum volume the chipped off wax is
recast into a solis sphere find thediameter of the sphere

Ankush Jain · Accepted Answer

Given, diameter of wax cylinder = 21 cm âˆ´ radius of wax cylinder $= \frac{21}{2} cm$ and Height of cylinder = 21 cm A cone of maximum volume can be chipped off only when the height and diameter of the cone is equivalent to height and diameter of cylinder Now, the wax remained after chipping off cone is recast to form a sphere

âˆ´ Volume of cylinder = Volume of cone + volume of sphere $π r^{2} h = \frac{1}{3} π r^{2} h + \frac{4}{3} π {r_{1}}^{3} [r = radius of cylinder = radius of cone = \frac{21}{2} cm] r^{2} h = \frac{1}{3} r^{2} h + \frac{4}{3} {r_{1}}^{3} [where r_{1} = radius of sphere]$ On putting respective values, we get $(\frac{21}{2})^{2} \times 21 = \frac{1}{3} \times (\frac{21}{2})^{2} \times 21 + \frac{4}{3} {r_{1}}^{3} (\frac{21}{2})^{2} \times 21 - \frac{1}{3} \times (\frac{21}{2})^{2} \times 21 = \frac{4}{3} {r_{1}}^{3} \Rightarrow \frac{4}{3} {r_{1}}^{3} = \frac{2}{3} \times (\frac{21}{2})^{2} \times 21 \Rightarrow {r_{1}}^{3} = \frac{2}{3} \times (\frac{21}{2})^{2} \times 21 \times \frac{3}{4} \Rightarrow {r_{1}}^{3} = \frac{21 \times 21 \times 21}{2 \times 2 \times 2} \Rightarrow r_{1} = \frac{21}{2} cm ∴ diameter of sphere = 2 r_{1} = 2 \times \frac{21}{2} = 21 cm$

wax cylinder of diameter 21cm and height 21 cm is chipped off and shaped to form a cone of maximum volume.the chipped off wax is recast into a solis sphere.find thediameter of the sphere..