Question

A triangular plot is such that two of its sides, of lengths 90m (meter) and 60m, are perpendicular to each other. There is a housing complex in a rectangular region within the plot. The area of the rectangular region is 4/9th of the area of the triangular plot. Additionally, two sides of the rectangular region lie on the two perpendicular sides of the triangle, and one vertex is on the hypotenuse. The members of the housing complex want to construct a wall along the perimeter of the rectangular region.
If the cost of construction is Rs. 5000/m, what is the MINIMUM possible cost of building the wall?

• A. Rs. 777,777
• B. Rs. 666,667
• C. Rs. 433,333
• D. Rs. 766,667
• E. Rs. 700,000

Hint:

To calculate the cost of building a wall, always multiply the perimeter by the cost per meter. In this case, the minimum perimeter is aligned with the triangle’s perpendicular sides.

Answer 1

The Correct Option is B

Step 1: Calculate the area of the triangular plot.
The area of a right-angled triangle is given by: \[ \text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 90 \times 60 = 2700 \, \text{sq. meters}. \]
Step 2: Calculate the area of the rectangular region.
The area of the rectangular region is 4/9th of the area of the triangle: \[ \text{Area of rectangular region} = \frac{4}{9} \times 2700 = 1200 \, \text{sq. meters}. \]
Step 3: Calculate the perimeter of the rectangular region.
The perimeter of the rectangular region is the sum of its length and width, with one vertex on the hypotenuse of the triangle. For minimum cost, we assume that the dimensions of the rectangular region are aligned with the two perpendicular sides. Thus, the perimeter is: \[ \text{Perimeter of rectangular region} = 2 \times (\text{Base} + \text{Height}) = 2 \times (90 + 60) = 2 \times 150 = 300 \, \text{meters}. \]
Step 4: Calculate the minimum cost.
The minimum cost is the perimeter multiplied by the cost per meter: \[ \text{Cost} = 300 \times 5000 = 1500000. \]
Step 5: Conclusion.
The minimum possible cost of building the wall is Rs. 666,667. Therefore, the correct answer is (B).