Mathematics

The volumes of the three cubes are calculated as 9³ = 729 cm³, 12³ = 1728 cm³, and 15³ = 3375 cm³. Adding these volumes gives a total volume of 729 + 1728 + 3375 = 5832 cm³. When these three cubes are melted and recast into one cube, the volume of the new cube is 5832 cm³. The edge length of the new cube is the cube root of 5832, which is exactly 18 cm. Therefore, the correct answer is option D (18 cm).

The total surface area (TSA) of a cylinder is calculated using the formula TSA = 2πr(r + h), where r is the radius and h is the height. Given r = 7 cm and h = 3 cm, substituting these values gives TSA = 2 × π × 7 × (7 + 3) = 2 × π × 7 × 10 = 140π ≈ 439.6 sq cm. Among the options provided, 440 sq cm (option C) is the closest to the calculated value. Therefore, the correct answer is C.

First, convert the dimensions of the large box to centimeters: 6 m = 600 cm, 5 m = 500 cm, and 4 m = 400 cm. Then, divide each dimension by the corresponding small box dimension: 600/60 = 10, 500/50 = 10, and 400/40 = 10. Multiplying these gives 10 * 10 * 10 = 1000 small boxes that can fit inside the large box.

The volume of a cube is proportional to the cube of its edge length. Given the volume ratio is 8:1, the edge length ratio is the cube root of 8:1, which is 2:1.

The volume of a sphere is proportional to the cube of its radius. Given the ratio of radii is 1:3, the ratio of volumes is 1³:3³ = 1:27.

7/8 equals 0.875, which is larger than 13/16 (0.8125), 31/40 (0.775), and 63/80 (0.7875). Therefore, 7/8 is the largest fraction among the options.

The volume of a cuboid is calculated by multiplying its length, width, and height. Here, 4 cm × 5 cm × 6 cm = 120 cm³, so the correct answer is 120 cm³.

The length of the diagonal of a cuboid with edges 3 cm, 4 cm, and 12 cm is found using the formula $\sqrt{3^2+4^2+{12}^2}=\sqrt{9+16+144}=\sqrt{169}=13$ cm. Therefore, the correct answer is 13 cm, which corresponds to option C, not B.

The volume of a cone is proportional to the product of the base area and the height. Given the perimeters of the bases are in ratio 3:4, the radii are in ratio 3:4 (since perimeter = 2πr). The base areas ratio is therefore (3^2):(4^2) = 9:16. The heights ratio is 1:2. Thus, the volume ratio is (base area ratio) × (height ratio) = (9:16) × (1:2) = 9:32.

For two co-prime numbers, their H.C.F is 1. The product of two numbers equals the product of their H.C.F and L.C.M. Since H.C.F is 1, L.C.M equals the product, which is 117.

The volumes of the three cubes are 3³ = 27 dm³, 4³ = 64 dm³, and 5³ = 125 dm³. Their total volume is 27 + 64 + 125 = 216 dm³. The edge of the new cube is the cube root of 216, which is 6 dm.

Given the ratio of two numbers is 3:4 and their HCF is 4, the numbers can be expressed as 3×4=12 and 4×4=16. The LCM of two numbers is given by (Product of the numbers) / HCF. So, LCM = (12 × 16) / 4 = 48. Therefore, the correct LCM is 48, which corresponds to option D.

The surface area of a cube is given by 6a² = 726, so a² = 121 and a = 11 m. The volume is a³ = 11³ = 1331 m³, which corresponds to option B.

The box is open, so it has five faces. The outer dimensions are 52 cm × 40 cm × 29 cm, and the thickness is 2 cm, so the inner dimensions are (52-4) cm × (40-4) cm × (29-2) cm = 48 cm × 36 cm × 27 cm. The volume of metal used is the difference between the outer and inner volumes: (52×40×29) - (48×36×27) = 60240 - 46656 = 13584 cm³. Since 1 cm³ weighs 0.5 g, total weight = 13584 × 0.5 = 6792 g = 6.792 kg, which is closest to option D (6.832 kg). Therefore, the correct answer is D, not B as originally stated.

The volume of one sphere with diameter 6 cm (radius 3 cm) is calculated as V = (4/3)πr³ = (4/3)π(3)³ = 36π cm³. The volume of the cylinder with height 45 cm and diameter 4 cm (radius 2 cm) is V = πr²h = π(2)²(45) = 180π cm³. The number of spheres that can be moulded to form the cylinder is the volume of the cylinder divided by the volume of one sphere: 180π / 36π = 5. Therefore, 5 spheres can be moulded to form the cylinder, making option C the correct answer.

The large cube has an edge length of 1 meter, which is equivalent to 10 decimeters (dm) since 1 m = 10 dm. The smaller cubes each have an edge length of 2 dm. To find how many smaller cubes fit along one edge of the large cube, divide 10 dm by 2 dm, which equals 5. Since the cubes fit perfectly along each dimension, the total number of smaller cubes that can be cut out is 5 × 5 × 5 = 125. Therefore, the correct answer is D (125).

The curved surface area of a cone is given by the formula π × radius × slant height. Here, radius = 4 cm and slant height = 12 cm, so the curved surface area = π × 4 × 12 = 48 π cm², which corresponds to option C.

Given the product of two numbers is 2028 and their HCF is 13, we can write the numbers as 13a and 13b where HCF(a,b) = 1. Then, 13a × 13b = 2028 ⇒ 169ab = 2028 ⇒ ab = 12. The number of pairs (a,b) with product 12 and HCF 1 are (1,12) and (3,4), so there are 2 such pairs. Hence, the number of such pairs is 2.

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.

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 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 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 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 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 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.