Quantitative Aptitude — Mensuration

Volume of cone = (1/3)πr²h = (1/3) × (22/7) × (7 cm)² × 24 cm = (1/3) × 22 × 7 × 24 cm³ = 22 × 7 × 8 cm³ = 1232 cm³.

Let the radius be 'r' and height 'h = 2r'. Volume of Cylinder (Vc) = πr²h = πr²(2r) = 2πr³ Volume of Cone (Vco) = (1/3)πr²h = (1/3)πr²(2r) = (2/3)πr³ Volume of Sphere (Vs) = (4/3)πr³ Ratio Vc : Vco : Vs = 2πr³ : (2/3)πr³ : (4/3)πr³ Multiplying by 3/(2πr³) gives 3 : 1 : 2.

The formula for the volume of a sphere is V = (4/3)πr³. This shows that the volume is directly proportional to the cube of the radius (r³). The chart values (4.19, 33.5, 113, 268, 524) show a rapid, non-linear increase, consistent with a cubic relationship.

Radius of capsule (r) = 3 mm. Length of cylindrical part (h) = 10 mm. Volume of cylindrical part = πr²h = π(3)²(10) = 90π mm³. Volume of two hemispheres = Volume of one sphere = (4/3)πr³ = (4/3)π(3)³ = (4/3)π(27) = 36π mm³. Total volume of capsule = Volume of cylindrical part + Volume of two hemispheres = 90π + 36π = 126π mm³.

From the pie chart, the cylindrical part contributes 55% of the total volume. Volume of cylindrical part = 55% of 1000 cm³ = (55/100) × 1000 cm³ = 550 cm³.

Volume of large sphere = (4/3)π(12)³ cm³. Volume of one small sphere = (4/3)π(2)³ cm³. Number of small spheres = (Volume of large sphere) / (Volume of one small sphere) = [(4/3)π(12)³] / [(4/3)π(2)³] = (12/2)³ = 6³ = 216.

Volume of cuboid = length × breadth × height = 4 cm × 4 cm × 2 cm = 32 cm³. When recast, the volume remains the same. Volume of sphere = (4/3)πR³. Given V_sphere = 32 cm³ and π = 3. (4/3) × 3 × R³ = 32 4R³ = 32 R³ = 8 R = 2 cm. Diameter = 2R = 2 × 2 = 4 cm.

Let the outer radius be R and inner radius be r. The width of the track is (R - r). The difference in circumferences is 2πR - 2πr = 2π(R - r) = 33. Substituting π = 22/7, we get 2 * (22/7) * (R - r) = 33. Therefore, 44/7 * (R - r) = 33, which gives (R - r) = (33 * 7) / 44 = 21/4 = 5 whole 1/4 m.

The total surface area of a cube is given by the formula 6 * (side squared). Here, the side of the cube is 2.6 cm. So, total surface area = 6 * (2.6^2) = 6 * 6.76 = 40.56 cm squared.

Volume of the metallic sphere = Volume of the cylindrical wire. Radius of the sphere = 3 cm. Thickness of the wire means its diameter = 4 mm, so its radius = 2 mm = 0.2 cm. Volume of sphere = (4/3) * π * (radius cubed) = (4/3) * π * (3^3) = 36π. Volume of wire = π * (radius squared) * height = π * (0.2^2) * h = 0.04πh. Equating both volumes: 36π = 0.04πh, which gives h = 36 / 0.04 = 900 cm = 9 m.

Area of a rectangle = length * breadth. Given, area = 453.6 m squared and length = 27 m. So, 27 * breadth = 453.6, which gives breadth = 453.6 / 27 = 16.8 m. The perimeter of a rectangle = 2 * (length + breadth) = 2 * (27 + 16.8) = 2 * 43.8 = 87.6 m.

Let the length, width, and height of the cuboid be 4x, 3x, and 5x respectively. A cuboid has 12 edges (4 lengths, 4 widths, 4 heights). The sum of all edges = 4 * (length + width + height) = 144. So, 4 * (4x + 3x + 5x) = 144 => 4 * 12x = 144 => 48x = 144 => x = 3. Therefore, length = 12 cm, width = 9 cm, and height = 15 cm. Total surface area of a cuboid = 2 * (length * width + width * height + height * length) = 2 * (12 * 9 + 9 * 15 + 15 * 12) = 2 * (108 + 135 + 180) = 2 * 423 = 846 cm squared.

Circumference of a circle = 2 * π * r = 308. Substituting π = 22/7, we get 2 * (22/7) * r = 308 => 44/7 * r = 308 => r = (308 * 7) / 44 = 49 m. Area of the circle = π * (radius squared) = (22/7) * 49 * 49 = 22 * 7 * 49 = 154 * 49 = 7546 m squared.

Outer radius (R) = 9 cm. Thickness = 1 cm, so inner radius (r) = 9 - 1 = 8 cm. Volume of the metal in a spherical shell = (4/3) * π * (R cubed - r cubed) = (4/3) * (22/7) * (9^3 - 8^3) = 88/21 * (729 - 512) = 88/21 * 217 = (88 * 31) / 3 = 2728 / 3 = 909 whole 1/3 cubic cm.

Radius (r) = 9 m, Height (h) = 12 m. Slant height (l) = square root of (r squared + h squared) = square root of (9 squared + 12 squared) = square root of (81 + 144) = square root of 225 = 15 m. The material needed for a conical tent corresponds to its curved surface area = π * r * l = π * 9 * 15 = 135π m squared. Cost of the material = Area * Rate = 135π * (100 / π) = 135 * 100 = Rupees 13,500.

Diameter of hemisphere = 12 cm, so radius (r) = 6 cm. Height of cone (h) = 8 cm. Slant height of cone (l) = square root of (6 squared + 8 squared) = square root of 100 = 10 cm. Total surface area of the compound object = Curved surface area of cone + Curved surface area of hemisphere = π * r * l + 2 * π * (r squared) = π * 6 * 10 + 2 * π * (6 squared) = 60π + 72π = 132π cm squared. Cost of painting = 132 * 25 = Rupees 3,300.

Length of the rectangle = 14 cm, Breadth = 10 cm. Total area of the rectangle = length * breadth = 14 * 10 = 140 cm squared. A semicircle is drawn on the length, so its diameter is 14 cm, meaning its radius (r) = 7 cm. Area of the semicircle = (π * r squared) / 2 = (22/7 * 7 * 7) / 2 = 77 cm squared. Area of the shaded region = Area of rectangle - Area of semicircle = 140 - 77 = 63 cm squared.

The area of the four side walls of a cube (lateral surface area) is given by 4 * (edge squared). Given, 4 * (edge squared) = 36. Therefore, edge squared = 36 / 4 = 9. Taking the square root, edge = square root of 9 = 3 cm.

Let the initial side of the cube be x. Original volume = x cubed. When the side is doubled, the new side becomes 2x. New volume = (2x) cubed = 8 * (x cubed). Therefore, the new volume becomes 8 times the original volume.

Volume of a hemisphere = (2/3) * π * (radius cubed). Given, (2/3) * (22/7) * (r cubed) = 19404. This gives 44/21 * (r cubed) = 19404 => r cubed = (19404 * 21) / 44 = 441 * 21 = 21 * 21 * 21. Taking the cube root, r = 21 cm.

Each interior angle of a regular polygon = 180 minus (each exterior angle). Given interior angle = 108 degrees, so each exterior angle = 180 - 108 = 72 degrees. The number of sides of a regular polygon = 360 divided by each exterior angle = 360 / 72 = 5 sides.

Let the common radius be r and common height be h. Volume of cylinder = π * (r squared) * h. Volume of cone = 1/3 * π * (r squared) * h. The ratio of their volumes = [π * (r squared) * h] divided by [1/3 * π * (r squared) * h] = 1 divided by (1/3) = 3 : 1.