Quantitative Aptitude

Mensuration

2,507 Questions
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Mensuration Questions

Multiple choice

A solid metallic right circular cylinder and a solid metallic right circular cone are given. The cylinder and cone both have same height $h$ and same base radii $r$. The two solids are melted together and recast into a solid cylinder of radius $\displaystyle \frac{1}{2} r $. Then the height of the cylinder is $\displaystyle \frac{16}{3} h$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of cylinder = pi*r^2*h. Volume of cone = (1/3)pi*r^2*h. Total volume = (4/3)*pi*r^2*h. New cylinder has radius r/2, so volume = pi(r/2)^2*H = pi*(r^2/4)*H. Equating volumes: (4/3)*pi*r^2*h = (1/4)*pi*r^2*H. H = (16/3)*h.

Multiple choice

A cone is made from a circular sector of radius 2 units. If total surface area of cone (including base) is $\displaystyle\pi $ sq. units, find the area of the sector.
  1. $\displaystyle \left ( 3-2\sqrt{2} \right )\pi$
  2. $\displaystyle \left ( 2\sqrt{2}-2\right )\pi$
  3. $\displaystyle \sqrt{2}\pi$
  4. $\displaystyle \sqrt{3}\pi$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be the radius of the sector (2) and R be the radius of the cone base. Arc length of sector = 2*pi*R = theta*r. If the sector radius is 2, the slant height of the cone is 2. Total surface area = pi*R^2 + pi*R*slant = pi*R^2 + 2*pi*R = pi. So R^2 + 2R - 1 = 0. R = (-2 + sqrt(4 + 4))/2 = sqrt(2) - 1. Area of sector = (1/2)r^2*theta = (1/2)*r(r*theta) = (1/2)2(2*pi*R) = 2*pi*R = 2*pi*(sqrt(2) - 1).

Multiple choice

Two solid cylinders are $12\;cm\;and\;18\;cm$ in height. Their diameters are $12\;cm\;and\;16\;cm$ respectively. Both the cylinders are melted and the material is moulded into a solid right circular cone of height $33\;cm$. Find its diameter.
  1. $28\;cm$
  2. $20\;cm$
  3. $14\;cm$
  4. $24\;cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Volume of cylinder 1 = pi * 6^2 * 12 = 432pi. Volume of cylinder 2 = pi * 8^2 * 18 = 1152pi. Total volume = 1584pi. Cone volume = (1/3) * pi * r^2 * 33 = 11 * pi * r^2. 11 * r^2 = 1584 => r^2 = 144 => r = 12. Diameter = 24 cm.

Multiple choice

Three solid spheres of radius $1$ cm, $6$ cm and $8$ cm, respectively are melted together and cast into a single sphere. The radius $r$ of this sphere will be
  1. $\displaystyle r< 9$ cm
  2. $r=9$ cm
  3. $\displaystyle 9$ cm $< r< 10$ cm
  4. $\displaystyle r> 10$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The volume of the new sphere equals the sum of the volumes of the three smaller spheres: (4/3)pi*r^3 = (4/3)pi*(1^3 + 6^3 + 8^3). Thus, r^3 = 1 + 216 + 512 = 729. Since 9^3 = 729, r = 9 cm.