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A heap of wheat is in the form of a cone of diameter $9\ m$ and height $3.5\ m$. Find its volume. How much canvas cloth is required to just cover the heap? (Use $\pi = 3.14$).
Given:
A heap of wheat is in the form of a cone of diameter $9\ m$ and height $3.5\ m$.
To do:
We have to find its volume and the canvas cloth required to just cover the heap.
Solution:
Diameter of the conical heap of wheat $= 9\ m$
This implies,
Radius $(r)=\frac{9}{2} \mathrm{~m}$
Height of the conical heap $(h)=3.5 \mathrm{~m}$
$=\frac{7}{2}\ m$
Volume of the heap $=\frac{1}{3} \pi r^{2} h$
$=\frac{1}{3} \times 3.14 \times \frac{9}{2} \times \frac{9}{2} \times \frac{7}{2}$
$=74.18 \mathrm{~m}^{3}$
We know that,
$l^2=r^2+h^2$
$\Rightarrow l=\sqrt{r^{2}+h^{2}}$
$=\sqrt{(\frac{9}{2})^{2}+(\frac{7}{2})^{2}}$
$=\sqrt{\frac{81}{4}+\frac{49}{4}}$
$=\sqrt{\frac{130}{4}}$
$=\frac{\sqrt{130}}{2}$
Curved surface area of the heap $=\pi r l$
$=3.14 \times \frac{\sqrt{130}}{2} \times \frac{9}{2}$
$=3.14 \times \frac{11.4}{2} \times \frac{9}{2}$
$=80.54 \mathrm{~cm}^{2}$