Question

(a) The base diameter and slant height of a wooden cone is $10$ centimetre each. What is the volume of this cone?
(b) If this cone is carved into a sphere of maximum size, find the volume of the sphere.

Answer 1

(a) Diameter of a cone $=10cm$
Thus, radius of a cone $=r=5cm$
Slant height of a cone $=h=10cm$
Volume of a cone $=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\times \pi \times 5\times 5\times 10=\frac{250}{3}\pi c{m}^3$

(b) Base diameter $=10cm$, hence $BC=5cm$.
Height $=10cm$
In $△PBC,PB=\sqrt{P{C}^2+B{C}^2}$
$=\sqrt{{10}^2+5^2}$
$=\sqrt{125}=5\sqrt{5}cm$
Let $O$ be the centre of the sphere and $r$ be the radius of the sphere.
In quadrilateral $ABCO,AB=BC=5cm$, since tangents from an external point on the same circle are equal.
In $△OAP,OP=10-r$,
$AO=r,AP=PB-AB=5\sqrt{5}-5$
$O{P}^2=O{A}^2+A{P}^2$
$\Rightarrow (10-r{)}^2=r^2+(5\sqrt{5}-5{)}^2$
$\Rightarrow 100+r^2-20r=r^2+125+25-50\sqrt{5}$
$\Rightarrow 20r=50\sqrt{5}-50$
$\Rightarrow 20r=50\times 1.236\Rightarrow r=\frac{6.18}{2}\Rightarrow r=3.09cm$
Volume of the sphere $=\frac{4}{3}\pi r^3=\frac{4}{3}\times 3.14\times (3.09{)}^3$
$=\frac{4}{3}\times 3.14\times 29.5=123.2c{m}^3$