Math: Surface Areas And Volume Practice Test!

Explanation
A solid cube with an edge length of 32 cm can be divided into smaller cubes with an edge length of 4 cm. To find the number of smaller cubes, we need to calculate the total volume of the larger cube and divide it by the volume of the smaller cube. The volume of the larger cube is calculated by multiplying the length of one side by itself three times (32 * 32 * 32 = 32768 cm³). The volume of the smaller cube is calculated by multiplying the length of one side by itself three times (4 * 4 * 4 = 64 cm³). Dividing the volume of the larger cube by the volume of the smaller cube gives us the number of smaller cubes that can be cut from the larger cube, which is 32768 / 64 = 512.

Explanation
A sector is the region between an arc and two radii, joining the center to the end points of the arc. It is a portion of a circle that is enclosed by two radii and an arc. The term "segment" refers to the region between an arc and a chord, not radii. A semicircle is half of a circle, so it does not fit the description given. Therefore, the correct answer is sector.

Explanation
The lateral surface area of a cube is the sum of the areas of all its faces except the top and bottom faces. Since a cube has 6 equal faces, each face has an area equal to the side length squared. Therefore, the lateral surface area is equal to 4 times the side length squared. If the lateral surface area is 1600 cm2, then 4 times the side length squared is equal to 1600. Dividing both sides by 4 gives us the side length squared, which is 400. Taking the square root of 400 gives us the side length, which is 20 cm.

Explanation
The volume of a cylinder is calculated by multiplying the base area by the height. In this case, the base area is given as 154 sq. cm and the height is 5 cm. Multiplying these values gives us 770 cubic cm, which is the volume of the cylinder.

Explanation
The given cuboid has dimensions of 3 cm, 2 cm, and 2 cm. To find how many cubes of 1 cm side can be cut out of it, we need to divide each dimension of the cuboid by the side length of the cube. Dividing 3 cm by 1 cm gives 3 cubes in the length dimension. Dividing 2 cm by 1 cm gives 2 cubes in the breadth dimension. Dividing 2 cm by 1 cm gives 2 cubes in the height dimension. Multiplying these values together, we get 3 x 2 x 2 = 12 cubes. Therefore, 12 cubes of 1 cm side can be cut out of the cuboid.

Explanation
The curved surface area of a hemisphere is given by the formula 2πr^2, where r is the radius of the hemisphere. We are given that the curved surface area is 77 cm^2. By substituting this value into the formula and solving for r, we find that the radius is 3.5 cm.

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

Explanation
When the radius of a cylinder is doubled and the height is halved, the curved surface area remains the same. This is because the curved surface area of a cylinder is given by the formula 2πrh, where r is the radius and h is the height. When the radius is doubled, the term 2πr is also doubled. However, when the height is halved, the term h is also halved. Therefore, the change in the radius cancels out the change in the height, resulting in the same curved surface area.

Explanation
The curved surface area of a hemisphere of diameter 2r can be calculated using the formula 2πr^2. This formula represents the area of the curved surface of a hemisphere, which is the surface that wraps around the sphere. The factor of 2 accounts for the fact that a hemisphere is half of a sphere. The π represents the mathematical constant pi, and r represents the radius of the hemisphere. Therefore, the correct answer is 2πr^2.

Explanation
The given information states that the breadth of the room is twice its height and half its length. Let's assume the height of the room is H, the length is L, and the breadth is B. According to the given information, we can write the following equations:

B = 2H (breadth is twice the height)
B = L/2 (breadth is half the length)

Also, the volume of the room is given as 1000 m3. The volume of a rectangular room is calculated by multiplying its length, breadth, and height. So, we can write the equation:

L * B * H = 1000

By substituting the values of B from the previous equations, we get:

L * (2H) * H = 1000
2LH^2 = 1000
LH^2 = 500

From the given options, the only dimensions that satisfy this equation are 20 m × 10 m × 5 m.

Explanation
The volume of a cylinder is directly proportional to the square of its radius and its height. Since the volumes of the two cylinders are equal, and the height of one cylinder is twice that of the other, the ratio of their radii can be found by taking the square root of the ratio of their heights. Therefore, the ratio of their radii is 1 : √2.

Explanation
The length of the longest pole that can be put in the room is determined by the diagonal of the room. Using the Pythagorean theorem, we can calculate the diagonal as the square root of the sum of the squares of the dimensions. In this case, the diagonal is the square root of (10^2 + 10^2 + 5^2) = sqrt(200) = 14.14 m. Therefore, the longest pole that can be put in the room is 14.14 m, which is closest to 15 m.

Explanation
When the height of a cone is doubled, its volume is directly proportional to the height. Since the volume of a cone is calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height, doubling the height will result in doubling the volume. Therefore, the volume is increased by 100%.

Explanation
The weight of water contained in the vessel can be calculated by finding the volume of the vessel and then multiplying it by the weight of 1 cubic decimeter of water. The volume of a conical vessel can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the radius is half of the diameter, so it is 60 cm. The height is given as 105 cm. Plugging these values into the formula, we get V = (1/3)π(60^2)(105) = 372,000 cm^3. Since 1 cubic decimeter is equal to 1000 cm^3, the volume is 372 decimeters cubed. Multiplying this by the weight of 1 cubic decimeter of water, 1.5 kg, we get 372 * 1.5 = 558 kg. Therefore, the weight of water contained in the vessel is 594 kg.

Explanation
The total area to be painted can be found by calculating the surface area of the hollow hemispherical vessel. The surface area of a hollow hemisphere consists of the curved surface area of the outer hemisphere and the curved surface area of the inner hemisphere. The curved surface area of a hemisphere is given by 2πr^2, where r is the radius. In this case, the radius of the outer hemisphere is 25/2 cm and the radius of the inner hemisphere is 24/2 cm. Thus, the total area to be painted is 2π(25/2)^2 - 2π(24/2)^2.