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Examples 26 to 30 of Mensuration

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(26) Three cubes of  sides 8 m, 6 m and  1 m are melted to form a new cube. Find the surface area of  the cube so formed.

Soln. Volume of  new cube   =  Sum of  volumes of  given cubes.

=  ( 83 + 63 + 13 )  =  729  m3.

∴  Edge of  new cube  =  \sqrt[3]{729}  =  9  m.

∴  Surface area of  this new cube  =  6 l2  =  6 × (9) 2  =  486 m2.

(27) If each edge of  a cube is increased by 25 %, find the percentage increase in its surface area.

Soln.

In area, the increment is 5625 from 10000

Means 56.25 %

(28) Two cubes have their volumes in the ratio 8 : 27. Find the ratio of  their surface areas.

Soln. Let  their edges be  a  and  b.  Then,

\frac{a^3}{b^3}  = \frac{8}{27}     or  \left(\frac{a}{b}\right)^3 = \left(\frac{2}{3}\right)^3  or  \frac{a}{b} = \frac{2}{3}.

Ratio of  their surface areas  =  \frac{6a^2}{6b^2} = \frac{a^2}{b^2}  =  \left(\frac{a}{b}\right)^2 = \frac{4}{9},  i.e.,  4 : 9.

(29) A cube of  edge 15 cm is immersed completely in a rectangular vessel containing water. If the dimensions of  the base of  vessel are  15 cm  9 cm, find the rise in water level.

Soln. Increase in volume  =  Volume of  the cube  =  (15 \times 15 \times 15) cm3

∴ Rise in water level  =  \frac{\mathrm{Volume}}{\mathrm{Area}} = \frac{15\times15\times15}{15\times9}  =  25  cm.

(30) How many bricks each measuring 25 cm  12 cm  5 cm  will be needed to construct a wall 8 m long, 6 m high and 20 cm thick ?

Soln.

Number of  bricks required

=    \frac{\mathrm{Volume\ of\ \ wall\ in\ c}\mathrm{m}^3}{\mathrm{Volume\ of\ \ 1\ \ brick\ in\ c}\mathrm{m}^3}

=    \frac{800\times600\times20}{25\times12\times5}

=    6400.

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