Ex.13.1 Q7 Surface Areas and Volumes - NCERT Maths Class 9

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Question

Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions

\(\begin{align}25 \,\rm{cm} \times 20 \,\rm{cm} \times 5 \,\rm{cm}\end{align}\) and the smaller of dimensions \(\begin{align}15 \,\rm{cm} \times 12\,\rm{cm} \times 5 \,\rm{cm}.\end{align}\) For all the overlaps, \(5 \%\) of the total surface area is required extra. If the cost of the cardboard is \( \rm{Rs.}\, 4\) for \(\begin{align}1000\,\rm{cm^2} \end{align}\), find the cost of cardboard required for supplying \(250\) boxes of each kind.

Text Solution

 

Reasoning:

A cuboid has six faces and the total surface area is sum of the area of the six faces. So, the cost for supplying \(250\) boxes of each kind will be the summation of surface area of boxes multiplied by cost per cm square.

What is the known?

(i) Dimensions of the smaller and bigger boxes.

(ii) Cost of the card board.

What is the unknown?

Cost of the card board required for \(250\) boxes of each kind.

Steps:

For bigger box:

\[\begin{align}&\text{length (l)} = 25\,\rm{cm}\\&\text{breadth (b)} = 20\,\rm{cm}\\&\text{height (h) }= 5\,\rm{cm} \end{align}\]

Total surface area

\[\begin{align}&= 2\,\,(lb + bh + hl)\\&= 2\,\,[25 \times 20 + 25 \times 20 + 5 \times 25]\\ &= 2\,\,[500 + 100 + 125]\\ &= 1450\,\rm{cm^2} \end{align}\]

Card board required for all the overlaps is \(5\%\) of their total surface area.

\[\begin{align}∴ \frac{5}{{100}} \times 1450 = 72.5\,\rm{cm} \end{align}\]

Net cardboard required for bigger box 

\[\begin{align}&= 1450 + 72.5\\&=1522.5\,\rm{cm^2} \end{align}\]

Card board required for \(250\) such boxes 

\[\begin{align}&= 1522.5 \times 250 \\&= 380625\,\rm{cm^2} \end{align}\]

Cost for 1000 \(\rm{m^2} = \,\,Rs.\,4\)

∴ Cost for \(380625\) \(\rm{cm^2}\)

\[\begin{align} &= \frac{4}{{1000}} \times \,\,380625\\&= \rm{Rs.}\,1522.5 \end{align}\]

For smaller box:

\[\begin{align}&\text{length (l) }= 15\,\rm{cm} \\&\text{breadth (b)} = 12\,\rm{cm}\\&\text{height (h)} = 5\,\rm{cm} \end{align}\]

Total surface area

\[\begin{align}&= 2(lb + bh + hl)\\&= 2[50 \times 12 + 12 \times 5 + 5 \times 15]\\&= 2[180 + 60 + 75]\\&= 630\,\rm{cm^2} \end{align}\]

Card board required for all the overlaps is \(5\%\) of their total surface area.

\[\begin{align}∴ \frac{5}{{100}} \times 630 = 31.5\,\,c{m^2} \end{align}\]

Net surface area of the smaller box 

\[\begin{align} &= 630 + 31.5\\ &= 661.5\,\rm{cm^2} \end{align}\]

Card board required for \(250\) such boxes

\[\begin{align} &= 661.5 \times 250 \\ &= 165375\,\rm{cm^2} \end{align}\]

Cost of the card board is \(\rm{Rs}. 4\) for \(\begin{align}1000\,\rm{cm^2} \end{align}\)

For \(165375\) \(\rm{cm^2}\) cost is

\[\begin{align}&= \frac{4}{{1000}} \times 16537\\&= \rm{Rs.}\,661.50 \end{align}\]

Cost of card board required for supplying \(250\) boxes of each kind

\[\begin{align}&= 1522.50 + 661.50\\&= Rs. 2184 \end{align}\]

  
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