All questions of Mensuration for CLAT Exam

The figure with the largest area is a circle with a radius of 6 units.

Understanding the Problem
To determine the distance traveled by the cart, we need to analyze the revolutions of the front and back wheels.
Wheel Circumference
- Front Wheel: 30 ft
- Back Wheel: 36 ft
Revolutions of Wheels
Let the number of revolutions made by the back wheel be \( x \).
Then, the front wheel has done \( x + 5 \) revolutions.
Distance Traveled Calculation
The distance traveled by each wheel can be calculated using the formula:
- Distance = Circumference × Number of Revolutions
For the back wheel:
- Distance (Back Wheel) = 36 ft × \( x \) = \( 36x \) ft
For the front wheel:
- Distance (Front Wheel) = 30 ft × \( (x + 5) \) = \( 30(x + 5) \) ft
Setting Distances Equal
Since the cart travels the same distance regardless of which wheel is considered, we set the distances equal:
\[ 36x = 30(x + 5) \]
Expanding the equation:
- \( 36x = 30x + 150 \)
Simplifying:
- \( 36x - 30x = 150 \)
- \( 6x = 150 \)
- \( x = 25 \)
Calculating Distance
Now, substituting \( x \) back into either distance equation:
Distance traveled by the back wheel:
- \( 36x = 36 \times 25 = 900 \) ft
Conclusion
Thus, the total distance traveled by the cart when the front wheel has done five more revolutions than the rear wheel is 900 ft. Therefore, the correct answer is option D.

Solution:
Let's draw a diagram to visualize the problem:



From the diagram, we can see that the ungrazed area is made up of four triangles, one in each corner of the square field. To find the area of each triangle, we need to first find the length of the diagonal of each square formed by the horses.

Using Pythagoras' theorem, we can find the diagonal of each square:

$ \text{Diagonal of square} = \sqrt{70^2 + 70^2} = \sqrt{2}\times 70 \approx 98.99\text{ m} $

Now we can find the area of each triangle:

$ \text{Area of triangle} = \frac{1}{2}(\text{length})\times (\text{height}) $

The height of each triangle is the radius of the circle formed by the horses, which is half the diagonal of the square:

$ \text{Height of triangle} = \frac{1}{2}(\text{diagonal of square}) = \frac{1}{2}\times \sqrt{2}\times 70 \approx 49.495\text{ m} $

The length of each triangle is the distance between two horses, which is the same as the length of the side of the square:

$ \text{Length of triangle} = \text{Side of square} = 70\text{ m} $

Therefore, the area of each triangle is:

$ \text{Area of triangle} = \frac{1}{2}\times 70\times 49.495 \approx 1717.8\text{ sq.m} $

Since there are four triangles, the total ungrazed area is:

$ \text{Total ungrazed area} = 4\times \text{Area of triangle} \approx 4\times 1717.8 \approx 6871.2\text{ sq.m} $

However, we need to subtract this from the total area of the square field, which is:

$ \text{Total area of square field} = 70^2 = 4900\text{ sq.m} $

Therefore, the area left ungrazed by the horses is:

$ \text{Area left ungrazed} = \text{Total area of square field} - \text{Total ungrazed area} \approx 4900 - 6871.2 \approx 1050\text{ sq.m} $

Therefore, the correct answer is option A, 1050 sq.m.