There is a rectangular tank of length 180 m and breadth 120 m in a circular field. If the area of the land portion of the field is 40000 m$^2$, what is the radius of the field?

### Concept & Formula The total area of the circular field comprises the area of the rectangular tank and the area of the remaining land portion. $$ \text{Total Area} = \text{Area of tank} + \text{Area of land} $$ $$ \text{Area of rectangle} = l \times w $$ $$ \text{Area of circle} = \pi r^2 $$ ### Step-by-Step Solution 1. **Given:** Length of tank = $180 \text{ m}$, Breadth = $120 \text{ m}$. Area of land portion = $40000 \text{ m}^2$. 2. **Find the area of the rectangular tank:** Area = $180 \times 120 = 21600 \text{ m}^2$. 3. **Find the total area of the circular field:** Total Area = Area of tank + Area of land Total Area = $21600 + 40000 = 61600 \text{ m}^2$. 4. **Find the radius of the field ($r$):** $\pi r^2 = 61600$ $\frac{22}{7} \times r^2 = 61600 \Rightarrow r^2 = 61600 \times \frac{7}{22}$ $r^2 = 2800 \times 7 = 19600$ $r = \sqrt{19600} = 140 \text{ m}$. ### Exam Strategy & Shortcut Once you get $r^2 = 19600$, recognize that $196$ is the square of $14$, and the two zeros add a zero to the square root, giving $140$ instantly. ### Common Pitfall Failing to add the area of the tank to the area of the land to get the total area of the circular field will lead to an incorrect calculation for the radius. ### Final Answer Therefore, the correct answer is **140 m**.