Question

A cow is tied up for grazing inside a rectangular field of dimensions $40\text{m}\times 36\text{m}$ in one corner of the field by a rope of length 14 m. Find the area of the field left ungrazed by the cow.

Answer 1

Hint: To find the area of the field left ungrazed by the cow, we have to subtract the area of the gazed section from the area of the field. The area of the field will be equal to the area of a rectangle with length as 40m and breadth as 36m which is given by the formula $A=lb$ . The area of the gazed section will be equal to the area of a sector which is given by the formula $A=\left( \dfrac{\theta }{360{}^\circ } \right)\pi {{r}^{2}}$ , where $\theta $ will be $90{}^\circ $ and r will be the length of the rope.

Complete step by step answer:
We are given a rectangular field of dimensions $40\text{m}\times 36\text{m}$ . Therefore, we can write the length of the field, $l=40\text{m}$ and the breadth, $b=36\text{m}$ . We are also given that in one corner of the field a cow is tied by a rope of length 14 m. Let us illustrate this situation.

We can see from the above figure that a cow tied with a rope gazes in a circle. Since the vow is tied in a corner, it can only gaze in the area of DEF which is the area of a sector. To find the area of the field left ungrazed by the cow, that is the shaded portion of the figure, we have to subtract the area of the gazed section from the area of the field. In other words, we have to subtract the area of the sector from the area of the rectangle.
$\Rightarrow \text{Area of the field left ungrazed by the cow}=\text{Area of the field}-\text{Area of the grazed section}$
Let us find the area of the field. We know that area of a rectangle is given by
$A=l\times b$
Therefore, area of the field,
$\begin{align}
  & \Rightarrow {{A}_{r}}=40\times 36 \\
 & \Rightarrow {{A}_{r}}=1440\text{ }{{\text{m}}^{2}}...\left( ii \right) \\
\end{align}$
Now, let us find the area of the gazed section. We know that area of a sector is given by
$A=\left( \dfrac{\theta }{360{}^\circ } \right)\pi {{r}^{2}}$
Where, $\theta $ is the angle of the sector and r is the radius of the circle.
We know that the angles in a rectangle are $90{}^\circ $ and the radius of the sector will be the length of the rope. Therefore, the area of the gazed section can be found as follows.
$\begin{align}
  & \Rightarrow {{A}_{g}}=\left( \dfrac{90{}^\circ }{360{}^\circ } \right)\pi {{\left( 14 \right)}^{2}} \\
 & \Rightarrow {{A}_{g}}=\dfrac{1}{4}\times \dfrac{22}{7}\times 14\times 14 \\
 & \Rightarrow {{A}_{g}}=154\text{ }{{\text{m}}^{2}}...\left( iii \right) \\
\end{align}$
Let us substitute (ii) and (iii) in (i).
$\begin{align}
  & \Rightarrow \text{Area of the field left ungrazed by the cow}=1440-154 \\
 & \Rightarrow \text{Area of the field left ungrazed by the cow}=1286\text{ }{{\text{m}}^{2}} \\
\end{align}$
Therefore, the area of the field left ungrazed by the cow is $1286\text{ }{{\text{m}}^{2}}$ .

Note: Students must learn thoroughly the formulas of area and perimeter of shapes. Students have a chance of making a mistake by writing the formula for the area of a rectangle as $2\left( l+b \right)$ . In the above solution, we have considered the formula for the sector to find the area of the gazed section. We can see from the figure and the property of the rectangle that the angle of the sector is $90{}^\circ $ . Therefore, we can consider the area of the gazed section as the area of a quarter circle, that is, this area will be $\dfrac{1}{4}$ of the area of a circle. We know that the area of a circle is given by $A=\pi {{r}^{2}}$ . Therefore, area of the gazed section,
$\begin{align}
  & \Rightarrow {{A}_{g}}=\dfrac{\pi {{r}^{2}}}{4} \\
 & \Rightarrow {{A}_{g}}=\dfrac{22\times 14\times 14}{4\times 7} \\
 & \Rightarrow {{A}_{g}}=154\text{ }{{\text{m}}^{2}} \\
\end{align}$
Students must never forget to write the units at the end.