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NCERT Exemplar · Class 9 Mathematics Surface Areas and Volumes

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EXERCISE 13.4 1–8 (part 4 of 4)

  1. Exercise 1

    A cylindrical tube opened at both the ends is made of iron sheet which is 2\displaystyle 2 cm thick. If the outer diameter is 16\displaystyle 16 cm and its length is 100\displaystyle 100 cm, find how many cubic centimeters of iron has been used in making the tube ?

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    NCERT’s answer
    $\displaystyle 8800 \mathrm{~cm}^{3}$
    The iron forms a hollow cylindrical shell of outer radius \(\displaystyle R\), inner radius \(\displaystyle r\) and length \(\displaystyle h\). \[R = 8 \text{ cm}, \qquad r = R-2 = 6 \text{ cm}, \qquad h = 100 \text{ cm} \] \[V = \pi(R^2-r^2)h = \frac{22}{7}(8^2-6^2)(100) \] \[V = \frac{22}{7}(28)(100) = 8800 \text{ cm}^3 \] Answer: \(\displaystyle 8800 \text{ cm}^3\)

  2. Exercise 2

    A semi-circular sheet of metal of diameter 28cm is bent to form an open conical cup. Find the capacity of the cup.

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    NCERT’s printed answer for this exercise does not match its own question. This working follows the question as printed.

    The semicircle's radius becomes the cone's slant height; its arc becomes the base circle. \[l = 14 \text{ cm} \] \[\pi l = 2\pi r \implies r = \frac{l}{2} = 7 \text{ cm} \] \[h = \sqrt{l^2-r^2} = \sqrt{196-49} = 7\sqrt3 \text{ cm} \] \[V = \frac13\pi r^2h = \frac13\cdot\frac{22}{7}\cdot49\cdot7\sqrt3 = \frac{1078\sqrt3}{3} \text{ cm}^3 \] Answer: \(\displaystyle \dfrac{1078\sqrt3}{3}\approx622.38 \text{ cm}^3\)NCERT prints: $\displaystyle 677.6$ cm³ — a misprint: that value needs h ≈ $\displaystyle 13.2$ cm, but l = $\displaystyle 14$ cm and r = $\displaystyle 7$ cm give h = √(l² − r²) = $\displaystyle 7$√$\displaystyle 3$ ≈ $\displaystyle 12.12$ cm.

  3. Exercise 3

    A cloth having an area of 165 m2\displaystyle 165 \mathrm{~m}^{2} is shaped into the form of a conical tent of radius 5\displaystyle 5 m (i) How many students can sit in the tent if a student, on an average, occupies 57m2\displaystyle \frac{5}{7} m^{2} on the ground? (ii) Find the volume of the cone.

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    NCERT’s printed answer for this exercise does not match its own question. This working follows the question as printed.

    Students sit on the tent's base; the cloth forms only its curved surface.
    (i)
    \[\text{Base area} = \pi r^2 = \frac{22}{7}(5)^2 = \frac{550}{7} \text{ m}^2 \]
    \[n = \frac{550/7}{5/7} = 110 \]
    (ii)
    \[\pi r l = 165 \implies \frac{22}{7}(5)l = 165 \implies l = 10.5 \text{ m} \]
    \[h = \sqrt{l^2-r^2} = \sqrt{110.25-25} = \sqrt{85.25} \approx 9.23 \text{ m} \]
    \[V = \frac13\pi r^2h = \frac13\left(\frac{550}{7}\right)(9.23) \approx 241.7 \text{ m}^3 \]
    Answer: (i) \(\displaystyle 110\) students (ii) \(\displaystyle V\approx241.7 \text{ m}^3\)
    NCERT prints: $\displaystyle 110$, $\displaystyle 241.7$ cm³ — a misprint: r and l are in metres, so the volume is $\displaystyle 241.7$ m³, not cm³.

  4. Exercise 4

    The water for a factory is stored in a hemispherical tank whose internal diameter is 14\displaystyle 14 m . The tank contains 50\displaystyle 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Calculate the volume of water pumped into the tank.

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    NCERT’s answer
    $\displaystyle 668.66 \mathrm{~m}^{3}$
    $\displaystyle 1$ kilolitre equals \(\displaystyle 1 \text{ m}^3\), so the tank already holds \(\displaystyle 50 \text{ m}^3\) of its hemispherical capacity. \[r = 7 \text{ m} \] \[\text{Capacity} = \frac23\pi r^3 = \frac23\cdot\frac{22}{7}\cdot343 = \frac{2156}{3} \text{ m}^3 \] \[\text{Water pumped} = \frac{2156}{3}-50 = \frac{2006}{3}\approx668.67 \text{ m}^3 \] Answer: \(\displaystyle \dfrac{2006}{3}\approx668.67 \text{ m}^3\)

  5. Exercise 5

    The volumes of the two spheres are in the ratio 64\displaystyle 64 : 27. Find the ratio of their surface areas.

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    NCERT’s answer
    $\displaystyle 16$ : $\displaystyle 9$
    \[\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 = \frac{64}{27} \implies \frac{r_1}{r_2} = \frac43 \] \[\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2 = \frac{16}{9} \] Answer: \(\displaystyle 16:9\)

  6. Exercise 6

    A cube of side 4\displaystyle 4 cm contains a sphere touching its sides. Find the volume of the gap in between.

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    NCERT’s answer
    $\displaystyle 30.48 \mathrm{~cm}^{3}$
    The sphere is inscribed touching all six faces, so its diameter equals the cube's edge. \[r = \frac{4}{2} = 2 \text{ cm} \] \[V_{\text{cube}} = 4^3 = 64 \text{ cm}^3 \] \[V_{\text{sphere}} = \frac43\pi r^3 = \frac43\cdot\frac{22}{7}\cdot8 = \frac{704}{21} \text{ cm}^3 \] \[\text{Gap} = 64-\frac{704}{21} = \frac{640}{21}\approx30.48 \text{ cm}^3 \] Answer: \(\displaystyle \dfrac{640}{21}\approx30.48 \text{ cm}^3\)

  7. Exercise 7

    A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height ?

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    NCERT’s answer
    $\displaystyle 50$%
    \[V_{\text{sphere}} = V_{\text{cylinder}} \] \[\frac{4}{3}\pi r^3 = \pi r^2 h \] \[h = \frac{4r}{3} \] Diameter of the cylinder, \(\displaystyle d = 2r \). \[d - h = 2r - \frac{4r}{3} = \frac{2r}{3} \] \[\frac{d-h}{h}\times 100 = \frac{2r/3}{4r/3}\times 100 = 50 \]Answer: the diameter exceeds the height by \(\displaystyle 50\%\).

  8. Exercise 8

    30\displaystyle 30 circular plates, each of radius 14\displaystyle 14 cm and thickness 3cm are placed one above the another to form a cylindrical solid. Find : (i) the total surface area (ii) volume of the cylinder so formed.

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    NCERT’s answer
    (i)
    $\displaystyle 9152 \mathrm{~cm}^{2}$ (ii) $\displaystyle 55440 \mathrm{~cm}^{3}$
    Height of the stack: \[h = 30 \times 3 = 90 \text{ cm} \] Radius \(\displaystyle r = 14 \) cm.Total surface area: \[S = 2\pi r (r + h) \] \[S = 2\times\frac{22}{7}\times 14\times(14+90) \] \[S = 9152 \text{ cm}^2 \]Volume: \[V = \pi r^2 h \] \[V = \frac{22}{7}\times 14^2\times 90 \] \[V = 55440 \text{ cm}^3 \]Answer: total surface area \(\displaystyle =9152\ \text{cm}^2\); volume \(\displaystyle =55440\ \text{cm}^3\).