Volume and Surface Area Problems and Solutions

• 8. A cylindrical tank of diameter 35 cm is full of water. If 11 litres of water is drawn off, the water level in the tank will drop by:

  1. \begin{aligned} 11\frac{3}{7} cm \end{aligned}
  2. \begin{aligned} 11\frac{2}{7} cm \end{aligned}
  3. \begin{aligned} 11\frac{1}{7} cm\end{aligned}
  4. \begin{aligned} 11 cm\end{aligned}

  Answer :

  Option A

  Explanation:

  Let the drop in the water level be h cm, then,
  
  \begin{aligned}
  \text{Volume of cylinder= }\pi r^2h \\
  => \frac{22}{7}*\frac{35}{2}*\frac{35}{2}*h = 11000 \\
  => h = \frac{11000*7*4}{22*35*35}cm\\
  = \frac{80}{7}cm\\
  = 11\frac{3}{7} cm
  \end{aligned}

• 9. A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off so as to make an open box. If the length of the square is 8 m, the volume of the box (in m cube) is:

  1. 4120 m cube
  2. 4140 m cube
  3. 5140 m cube
  4. 5120 m cube

  Answer :

  Option D

  Explanation:

  l = (48 - 16)m = 32 m, [because 8+8 = 16]
  b = (36 -16)m = 20 m,
  h = 8 m.
  
  Volume of the box = (32 x 20 x 8) m cube
  = 5120 m cube.

• 10. The cost of the paint is Rs. 36.50 per kg. If 1 kg of paint covers 16 square feet, how much will it cost to paint outside of a cube having 8 feet each side.

  1. Rs. 850
  2. Rs. 860
  3. Rs. 876
  4. Rs. 886

  Answer :

  Option C

  Explanation:

  We will first calculate the Surface area of cube, then we will calculate the quantity of paint required to get answer.
  Here we go,
  
  \begin{aligned}
  \text{Surface area =}6a^2 \\
  = 6 * 8^2 = 384 \text{sq feet} \\
  \text{Quantity required =}\frac{384}{16} \\
  = 24 kg\\
  \text{Cost of painting =} 36.50*24 \\
  = Rs. 876
  \end{aligned}

• 11. Two right circular cylinders of equal volumes have their heights in the ratio 1:2. Find the ratio of their radii.

  1. \begin{aligned} \sqrt{3}:1 \end{aligned}
  2. \begin{aligned} \sqrt{7}:1 \end{aligned}
  3. \begin{aligned} \sqrt{2}:1 \end{aligned}
  4. \begin{aligned} 2:1 \end{aligned}

  Answer :

  Option C

  Explanation:

  Let their heights be h and 2h and radii be r and R respectively then.
  
  \begin{aligned}
  \pi r^2h = \pi R^2(2h) \\
  => \frac{r^2}{R^2} = \frac{2h}{h} \\
  = \frac{2}{1} \\
  => \frac{r}{R} = \frac{\sqrt{2}}{1} \\
  => r:R = \sqrt{2}:1 \\
  \end{aligned}

• 12. 12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. Find the diameter of each sphere.

  1. 4 cm
  2. 6 cm
  3. 8 cm
  4. 10 cm

  Answer :

  Option A

  Explanation:

  In this type of question, just equate the two volumes to get the answer as,
  \begin{aligned}
  \text{Volume of cylinder =}\pi r^2h\\
  \text{Volume of sphere =} \frac{4}{3}\pi r^3\\
  => 12*\frac{4}{3}\pi r^3 = \pi r^2h \\
  => 12*\frac{4}{3}\pi r^3 = \pi *8*8*2 \\
  => r^3 = \frac{8*8*2*3}{12*4} \\
  => r^3 = 8 \\
  => r = 2 cm \\
  => \text{Diameter =}2*2 = 4 cm
  \end{aligned}

• 13. The slant height of a conical mountain is 2.5 km and the area of its base is 1.54 km square. The height of mountain is :

  1. 2.3 km
  2. 2.4 km
  3. 2.5 km
  4. 2.6 km

  Answer :

  Option B

  Explanation:

  Let the radius of the base be r km. Then,
  \begin{aligned}
  \pi r^2 = 1.54 \\
  r^2 = \frac{1.54*7}{22} = 0.49\\
  = 0.7 km \\
  \text{Now l=2.5 km, r = 0.7 km} \\
  h = \sqrt{2.5^2 - 0.7^2} km \\
  =\sqrt{6.25 - 0.49}\\
  =\sqrt{5.76} km \\
  = 2.4 km
  \end{aligned}

• 14. The radii of two cones are in ratio 2:1, their volumes are equal. Find the ratio of their heights.

  1. 1:4
  2. 1:3
  3. 1:2
  4. 1:5
  5. AC
  6. AF
  7. CF
  8. CE

  Answer :

  Option

  Explanation:

  Let their radii be 2x, x and their heights be h and H resp.
  Then,
  \begin{aligned}
  \text{Volume of cone =}\frac{1}{3}\pi r^2h \\
  \frac{\frac{1}{3}*\pi *{(2x)}^2*h}{\frac{1}{3}*\pi *{x}^2*H} \\
  => \frac{h}{H} = \frac{1}{4} \\
  => h:H = 1:4
  \end{aligned}