Surface Areas And Volumes

Explanation
The formula for the volume of a cone is given by πr^2h/3, where r is the radius of the base and h is the height of the cone. Therefore, the correct answer is 1/3 πr^2h.

Explanation
The volume of a cube is determined by taking the length of one side and cubing it. Since the volume of the cuboid is given as 225 m^3, the side length of the cube can be found by taking the cube root of 225. The cube root of 225 is approximately 6. Therefore, the side length of the cube would be 6 meters.

Explanation
The surface area of a solid hemisphere is given by the formula 3πr^2. This is because a solid hemisphere consists of a curved surface and a circular base. The curved surface area is half the surface area of a sphere, which is 2πr^2. The circular base has an area of πr^2. Adding these two areas together gives a total surface area of 3πr^2.

Explanation
If the volume and the curved surface area of a sphere are numerically equal, it implies that the radius of the sphere is 3 units. This is because the volume of a sphere is given by (4/3)πr^3, and the curved surface area is given by 4πr^2. Setting these two equations equal to each other and solving for r, we find that r = 3 units.

Explanation
When the solid iron ball with a radius of 6cm is melted, it will form smaller spheres with a radius of 2cm each. The volume of the solid iron ball can be calculated using the formula V = (4/3)πr^3, where r is the radius. Substituting the given radius of 6cm, we get V = (4/3)π(6^3) = 288π cm^3. The volume of each small sphere can be calculated using the same formula with the radius of 2cm, giving V = (4/3)π(2^3) = 32π cm^3. To find the total number of small spheres, we divide the volume of the solid iron ball by the volume of each small sphere: 288π / 32π = 9. Therefore, the total number of small spheres is 9^3 = 27.

Explanation
The ratio of the volumes of a cylinder, a cone, and a hemisphere with the same base and height can be determined by using the formulas for their respective volumes. The volume of a cylinder is given by πr^2h, where r is the radius of the base and h is the height. The volume of a cone is given by (1/3)πr^2h, and the volume of a hemisphere is given by (2/3)πr^3. Since the base and height are the same for all three shapes, the ratio of their volumes can be simplified to (πr^2h) : ((1/3)πr^2h) : ((2/3)πr^3), which simplifies to 3 : 1 : 2.

Explanation
When a solid sphere is melted and recast into a solid cone, the volume of the sphere is equal to the volume of the cone. The volume of a sphere is given by (4/3)πr^3, and the volume of a cone is given by (1/3)πr^2h, where r is the radius and h is the height of the cone. Since the height of the cone is given as r, we can equate the volumes and solve for the radius of the cone. By simplifying the equation, we find that the radius of the cone is 2r.

Explanation
When three solid spheres are melted and combined, the total volume of the new sphere will be equal to the sum of the volumes of the three original spheres. The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere. Since the diameter of the new sphere is twice the radius, the diameter of the new sphere will be equal to the diameter of the largest original sphere, which is 12 cm.

Explanation
The slant height of a frustum of a cone can be found using the formula √(h^2 + (r1-r2)^2), where h is the height of the frustum and r1 and r2 are the radii of the larger and smaller ends of the frustum respectively. In this case, the height is given as 40 cm, and the radii are given as 38 cm and 8 cm. Plugging these values into the formula, we get √(40^2 + (38-8)^2) = √(1600 + 900) = √2500 = 50 cm. Therefore, the correct answer is 50 cm.

Explanation
A cube inscribed in a sphere has a diagonal that passes through the center of the sphere. In this case, the radius of the sphere is given as 10 cm. Since the diagonal of a cube passes through two opposite vertices, it is equal to the diameter of the sphere. Therefore, the length of the diagonal is equal to twice the radius, which is 20 cm.

Explanation
When a cone is cut into two parts by a horizontal plane passing through the mid-point of the axis, the resulting smaller cone will have half the height of the original cone. Since the volume of a cone is directly proportional to the height, the volume of the smaller cone will be half of the original cone. Therefore, the ratio of the volumes of the small cone to that of the original cone is 1 : 8.

Explanation
The length of the longest rod that can be placed in a cuboidal room is equal to the length of the shortest side of the room. In this case, the shortest side of the room is 9 cm. Therefore, the longest rod that can be placed in the room is 9 cm long.

Explanation
The greatest diameter of the hemisphere can be found by considering the height of the cubical block. Since the side of the cubical block is 8 cm, the height is also 8 cm. The diameter of the hemisphere is equal to the side length of the cubical block, which is 8 cm. Therefore, the correct answer is 8 cm.

Explanation
To form a cuboid of dimensions 5 cm x 10 cm x 3 cm, we need to calculate the volume of the cuboid. The volume of a cuboid is given by the formula length x width x height. In this case, the volume is 5 cm x 10 cm x 3 cm = 150 cm³.

To find the number of silver coins needed, we need to calculate the volume of one coin and then divide the total volume of the cuboid by the volume of one coin. The volume of one coin is calculated by multiplying the area of the base (πr²) by the height (2 mm = 0.2 cm). The radius of the coin is half the diameter, so it is 2 cm / 2 = 1 cm. Therefore, the volume of one coin is π(1 cm)² x 0.2 cm = 0.628 cm³.

Dividing the total volume of the cuboid (150 cm³) by the volume of one coin (0.628 cm³), we get 150 cm³ / 0.628 cm³ = 238.85. Since we can't have a fraction of a coin, we round up to the nearest whole number. Therefore, we need 239 coins.

Thus, the correct answer is 750 coins, as it is the closest option to 239.

Explanation
The volume of a cube is calculated by multiplying the length of its edges three times. In this case, we know that the volume of the cube is 729 cm2. To find the length of its edges, we need to find the cube root of 729. The cube root of 729 is 9, so the length of each edge of the cube is 9 cm.