Mathematics

Given the perimeter of the rhombus is 68 cm, each side length is 68/4 = 17 cm. One diagonal is given as 16 cm, so half of this diagonal is 8 cm. Using the Pythagorean theorem on the right triangle formed by half the diagonals and the side, the half of the other diagonal is $\sqrt{17^2-8^2}=\sqrt{289-64}=\sqrt{225}=15$ cm. Therefore, the other diagonal is 30 cm. The area of the rhombus is half the product of the diagonals: $\frac{1}{2}\times 16\times 30=240$ cm². Hence, the correct answer is B (240 cm²).

The diagonal of a square relates to its side length by the formula diagonal = side × √2. Given the diagonal is 20 m, the side length is 20/√2 = 10√2 m. The area is side² = (10√2)² = 100 × 2 = 200 sq m. However, the calculation shows 200 sq m, which corresponds to option C, not D. Rechecking: diagonal = 20 m, side = 20/√2 = 14.142 m approx, area = 14.142² = 200 sq m approx. So the correct answer is C, not D. The original correct_answer was C, which is correct. The explanation was missing and is now provided. The topic is Geometry, difficulty is easy, and tags include square, diagonal, area, geometry.

If the base is increased by 10%, the new base is 1.1 times the original. Since the area remains unchanged, the altitude must decrease so that (new base) × (new altitude) = original area. Therefore, new altitude = original altitude / 1.1, which is approximately a 9 1/11% decrease.

Let the breadth be x meters. Then the length is 3x meters. The perimeter P = 2(length + breadth) = 2(3x + x) = 8x = 96, so x = 12. The area = length × breadth = 3x × x = 3 × 12 × 12 = 432 sq m.

Let the width be w and height be h. Given length = 15 m, width = 12 m. The sum of floor and ceiling areas = 2 × (length × width) = 2 × (15 × 12) = 360 m². The sum of the areas of the 4 walls = 2 × height × (length + width) = 2h(15 + 12) = 54h. Equate: 360 = 54h → h = 360/54 = 20/3 ≈ 6.67 m. Volume = length × width × height = 15 × 12 × 6.67 = 1200 m³. Thus, option B is correct.

The area grazed by the cow is a circle with area 9856 sq.m. Using the formula for the area of a circle, A = πr², we solve for the radius r: r = √(A/π) = √(9856/3.1416) ≈ 56 m. This means the length of the rope must be approximately 56 meters. Among the given options, 56 m corresponds to option A, making it the correct answer.

The area of a square is calculated by squaring the length of one side. Here, 15 m × 15 m = 225 sq m, making option A correct.

The cost of surrounding the land is Rs 3000 at Rs 7.50 per meter, so the perimeter is 3000 ÷ 7.50 = 400 meters. Let the length be 5x and breadth be 3x. The perimeter is 2(5x + 3x) = 16x = 400, so x = 25. The difference between length and breadth is 5x - 3x = 2x = 50 meters.

The diagonal of a square relates to its side length by the formula diagonal = side × √2. Given the diagonal AC = 5.2 cm, the side length = 5.2 / √2 ≈ 3.676 cm. The area = side² ≈ 3.676² = 13.52 sq.cm. However, the closest option to this calculation is B (13.52 sq.cm), so the correct answer should be B, not C. Upon rechecking, option B is correct and matches the calculation exactly.

The area of a trapezium is given by (1/2) × (sum of parallel sides) × height. Here, the parallel sides are 10 m and 6 m, and the area is 640 sq m. So, 640 = 0.5 × (10 + 6) × depth, which simplifies to 640 = 8 × depth, giving depth = 80 m. However, this calculation shows depth = 80 m, which matches option D, so the original answer D is correct. Rechecking the calculation: 0.5 × (10 + 6) = 8, then depth = 640 / 8 = 80 m. Therefore, the correct answer is D, not C. The explanation is updated accordingly.