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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}

  • 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}

  • 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.

  • 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}

  • 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}

  • 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}

  • 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