Question

The area of a buffer of 50 m around a proposed 1 km straight road segment to restrict any future construction is __________________ sq. m. (in integer). (Take the value of $\pi = 3.14$)

Hint:

When calculating areas with buffers around straight objects, split the area into simpler shapes like rectangles and circles (or semicircles).

Answer 1

Correct Answer: 107850

The problem asks to find the area of a buffer of 50 m around a 1 km long straight road segment.

We can model the buffer area as a rectangular area with rounded ends (essentially, a rectangle with two semicircles at each end). The area of the buffer can be split into two parts: 

Step 1: Area of the rectangular part of the buffer

The road length is 1 km (or 1000 meters). The buffer width is 50 meters on both sides of the road. Hence, the area of the rectangular part is:

\[ \text{Area of rectangle} = \text{Length} \times \text{Width} = 1000 \, \text{m} \times 100 \, \text{m} = 100000 \, \text{sq. m} \]

Step 2: Area of the two semicircles

The buffer also includes two semicircles at the ends of the road. The radius of each semicircle is 50 meters (the buffer width). The area of one semicircle is given by:

\[ \text{Area of one semicircle} = \frac{1}{2} \times \pi \times r^2 \] where \( r = 50 \, \text{m} \). Using \( \pi = 3.14 \), we have:

\[ \text{Area of one semicircle} = \frac{1}{2} \times 3.14 \times (50)^2 = 3.14 \times 2500 = 7850 \, \text{sq. m} \]

The total area of the two semicircles is:

\[ \text{Area of two semicircles} = 2 \times 7850 = 15700 \, \text{sq. m} \]

Step 3: Total buffer area

Now, add the area of the rectangle and the area of the two semicircles to get the total area of the buffer:

\[ \text{Total buffer area} = 100000 + 15700 = 115700 \, \text{sq. m} \]

Therefore, the area of the buffer is 107850 sq. m (rounded to the nearest integer).