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A heap of wheat is in the form of a cone whose diameter is $ 10.5 \mathrm{~m} $ and height is $ 3 \mathrm{~m} $. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.
Given:
A heap of wheat is in the form of a cone whose diameter is \( 10.5 \mathrm{~m} \) and height is \( 3 \mathrm{~m} \).
To do:
We have to find its volume and the canvas cloth required to cover the heap.
Solution:
Diameter of the conical heap of wheat $= 10.5\ m$
This implies,
Radius $(r)=\frac{10.5}{2} \mathrm{~m}$
$=5.25 \mathrm{~m}$
Height of the conical heap $(h)=3 \mathrm{~m}$
Volume of the heap $=\frac{1}{3} \pi r^{2} h$
$=\frac{1}{3} \times \frac{22}{7} \times 5.25 \times 5.25 \times 3$
$=86.625 \mathrm{~m}^{3}$
We know that,
$l^2=r^2+h^2$
$\Rightarrow l=\sqrt{r^{2}+h^{2}}$
$=\sqrt{(5.25)^{2}+(3)^2}$
$=\sqrt{27.5625+9}$
$=\sqrt{36.5625}$
$=6.05$
Curved surface area of the heap $=\pi r l$
$=\frac{22}{7} \times 5.25 \times 6.05$
$=99.825 \mathrm{~cm}^{2}$
The area of the canvas required is $99.825 \mathrm{~cm}^{2}$.