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 the cardboard required for supplying $250$ boxes of each kind.

Answer 1

Consider the problem, For bigger box
Given, Length of bigger box $=25cm$

Breadth of the bigger box $=20cm$

Height of bigger box $=5cm$

So total surface area of bigger box $=2\left(lb+lh+bh\right)$

$\begin{array}{c}=\left[2\left(25\times 20+25\times 5+20\times 5\right)\right]c{m}^2\\ =\left[2\left(500+125+100\right)\right]c{m}^2\\ =1450c{m}^2\end{array}$

Extra area required is equal to $5$ percent of surface area of bigger box

$\begin{array}{c}=\frac{5}{100}\times 1450\\ =72.5c{m}^2\end{array}$

Total area required for $1$ bigger box

$\begin{array}{c}=\text{Total, surface, area}+\text{Extra, area, required}\\ =\left(1450+72.5\right)c{m}^2\\ =1522.5c{m}^2\end{array}$

And total area required for $250$ box

$\begin{array}{c}=\text{Area, of one box}\times 250\\ =\left(1522.5\times 250\right)c{m}^2\\ =380625c{m}^2\end{array}$

For Smaller box, We have

Length of Smaller box $=15cm$

Breadth of the Smaller box $=12cm$

Height of Smaller box $=5cm$

So total surface area of Smaller box $=2\left(lb+lh+bh\right)$

$\begin{array}{c}=\left[2\left(15\times 12+15\times 5+12\times 5\right)\right]c{m}^2\\ =\left[2\left(180+75+60\right)\right]c{m}^2\\ =\left(2\times 315\right)c{m}^2\\ =630c{m}^2\end{array}$

Extra area required is equal to $5$ percent of surface area of Smaller box

$\begin{array}{c}=\frac{5}{100}\times 630\\ =31.5c{m}^2\end{array}$

Total area required for $1$ Smaller box $=\text{surface;Area}+\text{Extra;Area}$

$\begin{array}{c}=\left(630+31.5\right)c{m}^2\\ =661.5c{m}^2\end{array}$

Area of cardboard sheet required for $250$ such smaller boxes

$\begin{array}{c}=\text{Total, area}\times 250\\ =\left(250\times 661.5\right)c{m}^2\\ =165375\end{array}$

Total cardboard sheet required

$\begin{array}{c}=Areaof250big\text{box}+\text{Area, of}250\text{smaller, box}\\ =\left(380625+165375\right)c{m}^2\\ =546000c{m}^2\end{array}$

Cost of $1000c{m}^2$ cardboard sheet $=Rs4$

Cost of $1c{m}^2$ cardboard sheet $=Rs\frac{1}{1000}\times 4$

Cost of $546000c{m}^2$ card board sheet

$\begin{array}{c}=\left(\frac{546000\times 4}{1000}\right)\\ =Rs2184\end{array}$

Hence, the cost of card board sheet required for $250$ such boxes of each kind will be $Rs2184$