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

Answer 1

Since the cardboard boxes are cuboidal in shape, the total area of the cardboard is the same as the total surface area of the cuboid added to the overlap area that is, $5\%$ of the total surface area is required extra.

Hence, the area of each box can be obtained by adding $5\%$ of the total surface area to the total surface area of the cuboid.

For bigger boxes:

Dimensions of bigger boxes: $L=25cm,B=20cm,H=5cm$

Total surface area of the bigger box $(T.S.A)=2(lb+bh+lh)$

$\Rightarrow T.S.A=2\left\{\right(25cm\times 20cm)+(20cm\times 5cm)+(25cm\times 5cm\left)\right\}$

$\Rightarrow T.S.A=2\{500c{m}^2+100c{m}^2+125c{m}^2\}$

$\Rightarrow T.S.A=2\times 725c{m}^2$

$\Rightarrow T.S.A=1450c{m}^2$

With $5\%$ extra we get,

Requires area for each box $=1450c{m}^2+0.05\times 1450c{m}^2=1522.5c{m}^2$

For $250$ boxes , required are is $=1522.5c{m}^2\times 250=380625c{m}^2$ ---(1)

For smaller boxes:

Dimensions of smaller boxes: $l=15cm,b=12cm,h=5cm$

Total surface area of the bigger box $(t.s.a)=2(lb+bh+lh)$

$\Rightarrow (t.s.a)=2\left\{\right(15cm\times 12cm)+(12cm\times 5cm)+(15cm\times 5cm\left)\right\}$

$\Rightarrow (t.s.a)=2\{180c{m}^2+60c{m}^2+75c{m}^2\}$

$\Rightarrow (t.s.a)=2\times 315c{m}^2$

$\Rightarrow T.S.A=630c{m}^2$

With $5\%$ extra we get,

Required area for the box $=630c{m}^2+0.05\times 630c{m}^2=661.5c{m}^2$

For $250$ boxes , required are is $=661.5c{m}^2\times 250=165375c{m}^2$ ---(2)

So, Total $T.S.A$ of the boxes $=380625c{m}^2+165375c{m}^2=546000c{m}^2$

$Rs.4$ is charged or $1000c{m}^2$.

$\Rightarrow$ Total Cost $=\frac{546000c{m}^2}{1000c{m}^2}\times Rs.4=Rs.2184$