Mathematics

The side length of the square is the square root of its area, so side = √484 = 22 m. The perimeter of the square is 4 × 22 = 88 m. Since the perimeter of the circle equals the perimeter of the square, the circumference of the circle is also 88 m. Using the circumference formula 2πr = 88, solving for r gives r = 88 / (2π) = 14 m (using π ≈ 22/7). The diameter of the circle is twice the radius, so diameter = 2 × 14 = 28 m. Therefore, the correct answer is C (28 m).

To find the greatest number that divides 25, 73, and 97 leaving the same remainder, we find the greatest common divisor (GCD) of the differences between the numbers: 73 - 25 = 48, 97 - 73 = 24, and 97 - 25 = 72. The GCD of 48, 24, and 72 is 24. This means 24 is the greatest number that divides all three numbers leaving the same remainder. Therefore, the correct answer is 24 (option C).

The problem requires finding the least number n such that: n mod 2 = 1 n mod 3 = 2 n mod 4 = 3 n mod 5 = 4 n mod 6 = 5 and n mod 7 = 0. This implies that n + 1 is divisible by 2, 3, 4, 5, and 6. The least common multiple (LCM) of these numbers is 60, so n + 1 must be a multiple of 60. Also, n must be divisible by 7. So, n + 1 = 60k for some integer k, and n = 60k - 1. Since n is divisible by 7, 60k - 1 ≡ 0 (mod 7) => 60k ≡ 1 (mod 7). Reducing 60 mod 7 gives 60 ≡ 4 (mod 7), so 4k ≡ 1 (mod 7). The multiplicative inverse of 4 modulo 7 is 2, since 4*2=8 ≡ 1 (mod 7). Therefore, k ≡ 2 (mod 7). The smallest positive k satisfying this is k=2. Thus, n = 60*2 - 1 = 119. Checking 119: - 119 mod 2 = 1 - 119 mod 3 = 2 - 119 mod 4 = 3 - 119 mod 5 = 4 - 119 mod 6 = 5 - 119 mod 7 = 0 Hence, the least such number is 119. Among the options given, 119 (option B) is the correct answer.

The wire forms a square with area 484 sq.cm, so each side is 22 cm (since 22^2=484). The perimeter is 4×22=88 cm, which is the circumference of the circle formed by the same wire. Using C=2πr, radius r=88/(2π)=14 cm. The area of the circle is πr²=π×14²=196π ≈ 615.75 sq.cm, which matches option C. However, option B is given as 538 2/7 sq.cm, which is approximately 538.29, not matching the calculation. Therefore, option C (616 sq.cm) is the closest correct area. The original correct_answer was C, which is correct.

The floor area is 12.96 m × 3.84 m = 49.7664 m². To find the minimum number of whole square slabs, we need the largest square slab size that exactly divides both dimensions. The greatest common divisor (GCD) of 12.96 and 3.84 is 1.92 m. Each slab is 1.92 m × 1.92 m = 3.6864 m². Number of slabs = total area / slab area = 49.7664 / 3.6864 = 13.5 × 14 = 192 slabs.

The problem involves calculating the speed of water flow in a pipe that causes the water level in a tank to rise by a certain amount over a given time. Step 1: Calculate the volume increase in the tank: Volume = length × width × height increase = 7 m × 4 m × 0.045 m = 1.26 m³. Step 2: Convert the time to hours: 6 hours 18 minutes = 6 + 18/60 = 6.3 hours = 6.3 × 3600 = 22680 seconds. Step 3: Calculate the volumetric flow rate (Q) needed to raise the water level: Q = Volume / time = 1.26 m³ / 22680 s ≈ 5.56 × 10⁻⁵ m³/s. Step 4: Calculate the cross-sectional area of the pipe: Pipe width = 5 cm = 0.05 m, pipe depth = 4 cm = 0.04 m, Area = 0.05 m × 0.04 m = 0.002 m². Step 5: Calculate the velocity (v) of water in the pipe: v = Q / Area = (5.56 × 10⁻⁵ m³/s) / 0.002 m² = 0.0278 m/s. Step 6: Convert velocity to km/hr: 0.0278 m/s × 3600 s/hr = 100 m/hr = 0.1 km/hr. This velocity (0.1 km/hr) is much smaller than any of the options given (10 km/hr, 12 km/hr, 14 km/hr). Therefore, none of the options A, B, or C are correct. The correct answer is D (None of these). The original explanation incorrectly concluded the velocity should be 12 km/hr, but the calculations clearly show it is about 0.1 km/hr, which does not match any of the provided options except 'None of these'.

To find when A, B, and C meet again at the starting point, we calculate the least common multiple (LCM) of their times: 252, 308, and 198 seconds. The LCM is 2556 seconds, which converts to 42 minutes and 36 seconds. Therefore, they will meet again after 42 min 36 sec.

The area of the square is 6050 m². To find the side length, take the square root of the area: side = √6050 ≈ 77.78 m. The diagonal of a square is given by the formula diagonal = side × √2. Therefore, diagonal ≈ 77.78 × 1.414 ≈ 110 m. Among the options, 110 m corresponds to option A, making it the correct answer.

The circumference of the wheel is the distance covered in one revolution. Given total distance = 88 km = 88000 m and number of revolutions = 1000, the circumference = 88000 / 1000 = 88 m. The diameter is calculated by dividing the circumference by π (approximately 3.14), so diameter = 88 / 3.14 ≈ 28.03 m. This matches option C (28 m). Therefore, the correct answer is C, not B.

Increasing each side of a square by 50% means each side becomes 1.5 times the original length. The area scales by the square of the scale factor, so the new area is (1.5)^2 = 2.25 times the original. The ratio of the new area to the original area is therefore 9:4.