Question

A heap of rice is in the form of a cone of diameter 9 m and the height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap.

Answer 1

Hint: First we draw a diagram of a cone as given heap of rice is in the form of a cone. We have given diameter and height. We find the radius by using the relation $\text{Radius=}\dfrac{\text{Diameter}}{\text{2}}$ and then by using the formula of volume of the cone we find the volume of rice.
Volume of cone $=\dfrac{1}{3}\pi {{r}^{2}}h$.
Then, to obtain the required cloth, we calculate the surface area of the cone.

Complete step-by-step solution:
We have given that a heap of rice is in the form of a cone of diameter 9 m and the height 3.5 m.

We have to find the volume of the rice and area of canvas cloth required to just cover the heap.
To find the volume of rice, we use the formula of volume of a cone, which is given by
Volume of cone $=\dfrac{1}{3}\pi {{r}^{2}}h$.
We have given the diameter $=9m$
We know that $\text{Radius=}\dfrac{\text{Diameter}}{\text{2}}$
So, $\text{Radius=}\dfrac{9}{\text{2}}=4.5m$ and $h=3.5m$ as given in the question
Now substituting the values, we get
Volume of cone $=\dfrac{1}{3}\times \dfrac{22}{7}\times {{\left( 4.5 \right)}^{2}}\times 3.5$
Now, we solve further
Volume of cone
$\begin{align}
  & =\dfrac{1}{3}\times \dfrac{22}{7}\times 4.5\times 4.5\times 3.5 \\
 & =\dfrac{1559.25}{21} \\
 & =74.25\text{ }{{\text{m}}^{3}} \\
\end{align}$
So, the volume of rice is $74.25\text{ }{{\text{m}}^{3}}$.
Now, the cloth required to cover the heap of rice will be equal to the total surface area of the cone, which is $=\pi rl$.
Where,
 $\begin{align}
  & l=\sqrt{{{r}^{2}}+{{h}^{2}}} \\
 & l=\sqrt{{{4.5}^{2}}+{{3.5}^{2}}} \\
 & l=\sqrt{20.25+12.25} \\
 & l=\sqrt{32.5} \\
 & l=4.75m \\
\end{align}$
So, total surface area will be
$\begin{align}
  & =\dfrac{22}{7}\times 4.5\times 4.75 \\
 & =67.18\text{ }{{\text{m}}^{2}} \\
\end{align}$
So, $67.18\text{ }{{\text{m}}^{2}}$ canvas cloth is required to just cover the heap.

Note: In this question to find the volume of a cone, we need the value of radius but in question, we have given the value of diameter. If we directly use the value of diameter, it leads to an incorrect answer. So first we have to calculate the value of the radius of the cone by using the relation $\text{Radius=}\dfrac{\text{Diameter}}{\text{2}}$ and then put the values in the formula.