Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

Multiple choice

A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these have radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

  1. 8464 π

  2. 928 π

  3. 1026(1 + π )

  4. 1044(4 + π )
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To minimize the number of cylinders, the volume of each cylinder must be the greatest common divisor (GCD) of the metal volumes: 405, 783, and 351, which is 27 cc. With a radius of 3 cm, the height of each cylinder is h = 3/pi cm. The total number of cylinders is 57, and multiplying this by the surface area of a single cylinder, 18(1 + pi), gives a total surface area of 1026(1 + pi) sq cm.

Multiple choice

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

  1. 232

  2. 256

  3. 264

  4. 243
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The cone is cut at 18 cm from the base, so the top cone has height 27-18 = 9 cm. The ratio of heights is 9:27 = 1:3. The ratio of volumes is 1^3:3^3 = 1:27. Let total volume be V. Top volume is V/27, bottom frustum is 26V/27. Difference is 25V/27 = 225. V = 225 * 27 / 25 = 9 * 27 = 243.

Multiple choice

Directions: The following question consists of two quantities, Quantity I and Quantity II. You are to compare the two quantities and find the correct relationship between both the quantities. Quantity I: Area of a right triangle with base as 16 cm and hypotenuse as 4 cm more than base Quantity II: Area of a square with side 9 cm

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II, or No relation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Quantity I: Base = 16, Hypotenuse = 16+4 = 20. Height = sqrt(20^2 - 16^2) = sqrt(400 - 256) = sqrt(144) = 12. Area = 1/2 * 16 * 12 = 96. Quantity II: Area = 9^2 = 81. Since 96 > 81, Quantity I > Quantity II.

Multiple choice

The ratio of the total surface area to the lateral surface area of a cylindrical vessel is 27 : 20. The radius of the cylindrical vessel is 39 cm less than the height of the cylindrical vessel. The cylindrical vessel is filled up to 75% of its capacity with juice. The total juice from the cylindrical vessel is transferred into N conical vessels whose radius is 7 cm and height is 9 cm. What is the value of N?

  1. 125

  2. 135

  3. 164

  4. 165

  5. 170
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Total surface area of cylinder is 2*pi*r*(r+h) and lateral is 2*pi*r*h. Ratio (r+h)/h = 27/20. Given r = h - 39, substitute to get (2h-39)/h = 27/20. 40h - 780 = 27h, so 13h = 780, h = 60, r = 21. Volume = pi * 21^2 * 60. Juice = 0.75 * Volume. N = (0.75 * pi * 21^2 * 60) / (1/3 * pi * 7^2 * 9). N = (0.75 * 441 * 60) / (49 * 3) = 11907 / 147 = 135.

Multiple choice

In a scientific investigation, a rectangular container with base dimensions 150 centimeters by 120 centimeters is used. Inside this container, four identical circular lenses and two identical square lenses are arranged on the base. The lenses are positioned in a non-overlapping manner, meaning they do not overlap each other. Each circular lens has a radius of 7 centimeters, and each square lens has a side length of 12 centimeters. What percentage of the container's base area is not covered by the lenses?

  1. 62.5%

  2. 57.6%

  3. 51.3%

  4. 49.8%

  5. 45.1%
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

What should be the internal radius of a cylindrical metallic water pipe if the volume of metal used in the pipe is 1,672 cubic units, ensuring an adequate flow rate for the household's water needs, given that the external radius is 1 unit more than the internal radius and the length of the pipe is 28 units?

  1. 4 units

  2. 4.5 units

  3. 8 units

  4. 9 units

  5. 18 units
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Volume of metal = pi * h * (R^2 - r^2) = 1672. Here R = r+1, h=28. So 1672 = (22/7) * 28 * ((r+1)^2 - r^2). 1672 = 88 * (2r+1). 19 = 2r+1, so 2r=18, r=9.

Multiple choice

Tom possesses two cylinders which he intends to adorn and place in his lawn. In pursuit of this, he opts to envelop the curved surface areas of these cylinders with decorative material priced at $0.5 per square inch. The first cylinder has a radius of 35 inches and a height of 20 inches, while the second cylinder has a radius of 28 inches and a height of 35 inches. What is the total expenditure required?
  1. $12,260
  2. $10,560
  3. $6,280
  4. $5,280
  5. $4,200
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

CSA = 2*pi*r*h. Cylinder 1: 2*pi*35*20 = 1400*pi. Cylinder 2: 2*pi*28*35 = 1960*pi. Total area = 3360*pi. Using pi = 22/7, area = 3360 * 22/7 = 480 * 22 = 10,560 sq inches. Cost = 10,560 * 0.5 = 5,280.

Multiple choice

A dairy farm produces rectangular milk cartons for packaging milk. The dimensions of a carton are in the ratio of 5 : 4 : 3 (length : width : height). If the volume of one such carton is 480 cubic decimeters (dm3), determine the cost of laminating the entire outer surface of the carton at the rate of 3 cents per square decimeter (dm2).
  1. $8.90
  2. $10.28
  3. $11.28
  4. $13.76
  5. $14.36
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let dimensions be 5x, 4x, 3x. Volume = 60x^3 = 480, so x^3 = 8, x = 2. Dimensions are 10, 8, 6. Surface area = 2(10*8 + 8*6 + 6*10) = 2(80 + 48 + 60) = 2(188) = 376 dm^2. Cost = 376 * 0.03 = 11.28.

Multiple choice

A cylindrical tank full of oil, with the ratio of its radius to its height being 7 : 4, is used to fill 24 identical cuboidal containers with dimensions 11 × 9 × 7. What is the sum of the radius and the height of the cylindrical tank?

  1. 22 units

  2. 28 units

  3. 33 units

  4. 42 units

  5. 48 units
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of 24 containers = 24 * 11 * 9 * 7 = 16632. Cylinder volume = pi * r^2 * h. Given r/h = 7/4, let r=7k, h=4k. Volume = pi * (49k^2) * (4k) = 196 * pi * k^3. Assuming pi=22/7, 196 * (22/7) * k^3 = 16632 -> 616 * k^3 = 16632 -> k^3 = 27 -> k=3. Radius = 21, height = 12. Sum = 33.

Multiple choice

A cylindrical water tank in the backyard of a house collects rainwater for the garden. If the homeowner decides to increase the radius of the tank by 29%, while keeping the height constant, what percent increase will occur in the volume of water the tank can hold?

  1. 21.41%

  2. 29%

  3. 41.41%

  4. 66.41%

  5. 81.81%
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The volume of a cylinder is V = pi * r^2 * h. If the radius increases by 29%, the new radius is 1.29r, so the new volume is pi * (1.29r)^2 * h = 1.6641 * pi * r^2 * h. This represents a 66.41% increase.

Multiple choice

Three solid lead spheres of diameters 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres?

  1. 25%

  2. 26%

  3. 27%

  4. 28%
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Sum of surface areas = 4*pi*(3^2 + 4^2 + 5^2) = 4*pi*(9+16+25) = 200*pi. New radius R^3 = 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216, so R = 6. New area = 4*pi*(6^2) = 144*pi. Diminution = (200-144)/200 = 56/200 = 28%.

Multiple choice

A solid sphere of radius 3 cm is melted to form a hollow cylinder of height 4 cm and external diameter 10 cm. What is the thickness of the cylinder?

  1. 0.42 cm

  2. 0.46 cm

  3. 0.50 cm

  4. 1.00 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The volume of the sphere is (4/3) * pi * 3^3 = 36 * pi. The volume of the hollow cylinder is pi * (R^2 - r^2) * h, where R = 5 cm (external radius) and h = 4 cm. Thus, 36 * pi = pi * (25 - r^2) * 4, which simplifies to 9 = 25 - r^2, so r^2 = 16 and r = 4 cm. The thickness is R - r = 5 - 4 = 1 cm.