Mathematics

The longest rod that can fit inside a cube is the space diagonal, calculated as side × √3. For a cube with side 4 m, the diagonal is 4 × √3 ≈ 6.928 m. Among the options, 6.928 m corresponds to option C, so option C is correct, not B.

The surface area of a cube is proportional to the square of its side length. Given the side lengths ratio 3:1, the surface area ratio is (3^2):(1^2) = 9:1.

Let the two numbers be 37a and 37b, where 37 is their HCF and a and b are coprime. Their product is given as 4107, so (37a)(37b) = 4107, which simplifies to 1369ab = 4107, giving ab = 3. Since a and b are coprime and multiply to 3, the possible pairs are (1, 3) or (3, 1). Thus, the two numbers are 37 × 1 = 37 and 37 × 3 = 111. The greater number is therefore 111, corresponding to option C.

The surface area of a sphere is proportional to the square of its radius, and the volume is proportional to the cube of its radius. Given the surface area ratio 1:4, the ratio of the radii is √1:√4 = 1:2. Therefore, the volume ratio is 1³:2³ = 1:8.

The volume of the large wooden box is 8m × 7m × 6m = 336 cubic meters. Converting to cubic centimeters (1m = 100cm), the volume is (800cm) × (700cm) × (600cm) = 336,000,000 cubic centimeters. Each small box has a volume of 8cm × 7cm × 6cm = 336 cubic centimeters. Dividing the total volume by the small box volume gives 336,000,000 ÷ 336 = 1,000,000 boxes. However, since the dimensions of the large box are exactly 100 times the small box in each dimension (800cm/8cm=100, 700cm/7cm=100, 600cm/6cm=100), the number of small boxes that fit is 100 × 100 × 100 = 1,000,000. Therefore, the correct answer is B (1,000,000). The original answer was B, which is correct. The explanation was missing and has been added. The difficulty is easy, and relevant tags have been added.

The volume of earth dug out from the tank is calculated as length × width × depth = 3 m × 2 m × 1.5 m = 9 m³. The area of the field is length × width = 20 m × 14 m = 280 m². When the earth is spread evenly over the field, the rise in the level of the field is volume ÷ area = 9 m³ ÷ 280 m² = 0.03214 m = 3.214 cm. Among the given options, 2.98 cm (option C) is the closest to the calculated value, so it is the correct answer.

The volume of a sphere is given by the formula V = (4/3)πr³. If the radius is doubled, the new volume becomes V' = (4/3)π(2r)³ = (4/3)π(8r³) = 8V. This means the volume increases by a factor of 8, which is 700% more than the original volume (since the original volume counts as 100%, the increase is 8 times minus 1 times the original volume, i.e., 700%). Therefore, the volume increases by 700%, corresponding to option C.

The volume of a cylinder is calculated using the formula V = πr²h. Given the diameter is 6 cm, the radius r = 6/2 = 3 cm. The height h is 5 cm. Substituting these values, the volume V = π × 3² × 5 = 45π cm³. The unit cc (cubic centimeters) is equivalent to cm³, so the volume is 45π cc. Therefore, the correct answer is option B: 45 π cc.

The number of divisors for each option is calculated as follows: - 99 = 3^2 * 11^1, number of divisors = (2+1)*(1+1) = 3*2 = 6 - 101 is prime, number of divisors = 2 - 176 = 2^4 * 11^1, number of divisors = (4+1)*(1+1) = 5*2 = 10 - 182 = 2^1 * 7^1 * 13^1, number of divisors = (1+1)*(1+1)*(1+1) = 2*2*2 = 8 Therefore, 176 has the greatest number of divisors (10). The original answer (182) was incorrect. The correct answer is option C (176).

The greatest number that exactly divides 35, 91, and 840 is their greatest common divisor (GCD). Calculating the GCD of 35, 91, and 840 gives 7, which is option C.