Volume and Surface Area Questions Answers

• 15. The curved surface of a right circular cone of height 15 cm and base diameter 16 cm is:

  1. \begin{aligned} 116 \pi cm^2 \end{aligned}
  2. \begin{aligned} 122 \pi cm^2 \end{aligned}
  3. \begin{aligned} 124 \pi cm^2 \end{aligned}
  4. \begin{aligned} 136 \pi cm^2 \end{aligned}

  Answer And Explanation

  Answer: Option D

  Explanation:

  \begin{aligned}
  \text{Curved surface area of cone=}\pi rl\\
  l = \sqrt{r^2+h^2} \\
  l = \sqrt{8^2+15^2} = 17cm \\
  
  \text{Curved surface area =}\pi rl\\
  = \pi *8*17 = 136 \pi cm^2
  \end{aligned}

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• 16. If a right circular cone of height 24 cm has a volume of 1232 cm cube, then the area of its curved surface is :

  1. \begin{aligned} 450 cm^2 \end{aligned}
  2. \begin{aligned} 550 cm^2 \end{aligned}
  3. \begin{aligned} 650 cm^2 \end{aligned}
  4. \begin{aligned} 750 cm^2 \end{aligned}

  Answer And Explanation

  Answer: Option B

  Explanation:

  Volume is given, we can calculate the radius from it, then by calculating slant height, we can get curved surface area.
  \begin{aligned}
  \frac{1}{3}*\pi *r^2*h = 1232 \\
  \frac{1}{3}*\frac{22}{7}*r^2*24 = 1232 \\
  r^2 = \frac{1232*7*3}{22*24} = 49 \\
  r = 7 \\
  \text{Now, r = 7cm and h = 24 cm } \\
  l = \sqrt{r^2+h^2} \\
  = \sqrt{7^2+24^2} = 25cm \\
  \text{Curved surface area =}\pi rl\\
  = \frac{22}{7}*7*25 = 550 cm^2
  \end{aligned}

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• 17. 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 And Explanation

  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.

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• 18. The maximum length of a pencil that can he kept is a rectangular box of dimensions 8 cm x 6 cm x 2 cm, is

  1. \begin{aligned} 2\sqrt{17} \end{aligned}
  2. \begin{aligned} 2\sqrt{16} \end{aligned}
  3. \begin{aligned} 2\sqrt{26} \end{aligned}
  4. \begin{aligned} 2\sqrt{24} \end{aligned}

  Answer And Explanation

  Answer: Option C

  Explanation:

  In this question we need to calculate the diagonal of cuboid,
  which is =
  \begin{aligned}
  \sqrt{l^2+b^2+h^2} \\
  = \sqrt{8^2+6^2+2^2} \\
  = \sqrt{104} \\
  = 2\sqrt{26}
  \end{aligned}

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• 19. 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 And Explanation

  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}

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• 20. 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

  Answer And Explanation

  Answer: Option A

  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}

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• 21. The volume of the largest right circular cone that can be cut out of a cube of edge 7 cm is:

  1. \begin{aligned} 79.8 cm^3 \end{aligned}
  2. \begin{aligned} 79.4 cm^3 \end{aligned}
  3. \begin{aligned} 89.8 cm^3 \end{aligned}
  4. \begin{aligned} 89.4 cm^3 \end{aligned}

  Answer And Explanation

  Answer: Option C

  Explanation:

  Volume of the largest cone = Volume of the cone with diameter of base 7 and height 7 cm
  
  \begin{aligned}
  \text{Volume of cone =}\frac{1}{3}\pi r^2h \\
  = \frac{1}{3}*\frac{22}{7}*3.5*3.5*7 \\
  = \frac{269.5}{3}cm^3 \\
  = 89.8 cm^3
  \end{aligned}
  
  Note: radius is taken as 3.5, as diameter is 7 cm

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