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

What is the volume of the hemisphere with radius $21$ cm?
  1. $16404$ $cm^3$
  2. $17404$ $cm^3$
  3. $18404$ $cm^3$
  4. $19404$ $cm^3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The volume of a hemisphere is given by (2/3) * pi * r^3. With r = 21, the volume is (2/3) * (22/7) * 21 * 21 * 21 = 2 * 22 * 3 * 21 * 21 / 3 = 44 * 21 * 21 = 19404 cubic cm.

Multiple choice

Small pebbles are continuously being added to a graduated cylindrical container containing water. The total volume of water and pebbles can be represented by the equation $V=24\pi +x\left(\displaystyle\frac{4}{3}\pi r^3\right)$, if the pebbles are considered to be spheres of radius $r$. If the volume of the graduated cylinder is $96\pi$ cubic centimetres, then, what is the maximum number of pebbles with a radius of $3$ centimetres that can be added without the volume of the fluid exceeding that of the graduated cylinder?
  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The total volume is 96pi. Initial water is 24pi. Available space = 96pi - 24pi = 72pi. Volume of one pebble = (4/3) * pi * 3^3 = (4/3) * pi * 27 = 36pi. Number of pebbles = 72pi / 36pi = 2.

Multiple choice

When a right triangle of area $4$ is rotated $360^o$ about its longer leg, the solid that results has a volume of $16$. Calculate the volume of the solid that results when the same right triangle is rotated about its shorter leg.
  1. $4.39$
  2. $8.77$
  3. $16.93$
  4. $35.09$
  5. $50.27$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the legs be a and b. Area = 0.5*a*b = 4, so ab = 8. Rotating about 'a' gives a cone with radius b and height a: Volume = (1/3)pi*b^2*a = 16. Thus, (1/3)*pi*b(ab) = 16 => (1/3)pi*b*8 = 16 => b = 6/pi. Then a = 8/b = 8/(6/pi) = 4*pi/3. Rotating about 'b' gives volume = (1/3)*pi*a^2*b = (1/3)*pi*a(ab) = (1/3)pi(4*pi/3)*8 = 32*pi^2/9 approx 35.09.

Multiple choice

Figure shows a semicircle that is the graph of the equation $y = \sqrt {6x - x^{2}}$. If the semicircle is rotated $360^{\circ}$ about the $x$-axis, calculate the volume of the sphere that is created.
  1. $6\pi$
  2. $12\pi$
  3. $18\pi$
  4. $24\pi$
  5. $36\pi$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

The equation y = sqrt(6x - x^2) is y^2 = 6x - x^2, or (x-3)^2 + y^2 = 9. This is a semicircle with radius 3 centered at (3, 0). Rotating this about the x-axis creates a sphere of radius 3. Volume = (4/3) * pi * r^3 = (4/3) * pi * 27 = 36 * pi.

Multiple choice

A formula for the surface area $\left(A\right)$ of the rectangular solid shown below is $A=2lw+2lh+2wh$ where $l$ represents length; $w$, width; and $h$, height. By doubling each of the dimensions ($l, w$, and $h$), the surface area will be multiplied by what factor?
  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If dimensions l, w, h are doubled to 2l, 2w, 2h, the new surface area is 2(2l)(2w) + 2(2l)(2h) + 2(2w)(2h) = 8lw + 8lh + 8wh = 4(2lw + 2lh + 2wh). The area is multiplied by 4.

Multiple choice

You can find the volume of an irregularly shaped solid object by completely submerging it in water and calculating the volume of water the object displaces. You completely submerge a solid object in a rectangular tank that has a base $40$ centimeters by $30$ centimeters and is filled with water to a depth of $20$ centimeters. The object sinks to the bottom, and the water level goes up $0.25$ centimeters. What is the volume, in cubic centimeters, of the object?
  1. $300$
  2. $240$
  3. $200$
  4. $150$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of the displaced water is equal to the volume of the object. Volume = base area * change in height = (40 * 30) * 0.25 = 1200 * 0.25 = 300 cubic centimeters.

Multiple choice

The sum of radius of the base and the height of a solid cylinder is $37 m$. If the total surface area of the cylinder be $1628{ m }^{ 2 }$, find the volume.
  1. $4620 cu. m$
  2. $4630 cu. m$
  3. $4520 cu. m$
  4. $4830 cu. m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let r be the radius and h be the height. We have r + h = 37 and 2*pi*r*(r + h) = 1628. Substituting 37 for (r + h), we get 2*pi*r*37 = 1628, so r = 1628 / (74 * pi) which is approximately 7. The volume is pi*r^2*h. Solving the system yields r = 7 and h = 30, so volume = pi * 49 * 30 = 1470 * pi, which is approximately 4620.