Question

Determine how many cuboidal boxes of size $20cm \times 10cm \times 5cm$ can be made from a cardboard of size $1.4m \times 1m$

Answer 1

Hint: To get the count of cuboidal boxes which can be made from the cardboard, Divide the area of the cardboard by the total surface area of each cuboidal box.
Formulas used:
Area of the cardboard (rectangle) is $l \times b$, where l is the length and b is the breadth.
Total surface area of a cuboid (box) is $2\left( {lb + bh + lh} \right)$, where l is the length, b is the breadth and h is the height of the cuboid.

Complete step-by-step answer:

We are given that a cuboidal box is of size $20cm \times 10cm \times 5cm$ and a cardboard sheet is of size $1.4m \times 1m$
We have to find the number of cuboidal boxes that can be made from the cardboard sheet.
Firstly, we have to find the area of the cardboard sheet.
Length of the sheet is 1.4m and breadth is 1m.
We already know that one metre is equal to 100 centimetres.
$
  1.4m = 1.4 \times 100cm = 140cm \\
  1m = 1 \times 100cm = 100cm \\
 $
Area of the cardboard is $l \times b$
$
  l = 140cm,b = 100cm \\
  \to Area = l \times b \\
  \to 140 \times 100 \\
  \to 14000c{m^2} \\
 \therefore Area = 14 \times {10^3}c{m^2} \\
 $
Now, we have to calculate the total surface area of a cuboidal box.
Total surface area of a cubical box is $2\left( {lb + bh + lh} \right)$, where l is the length, b is the breadth and h is the height of the cuboidal box.
Length, breadth and height of the cuboidal box are 20cm, 10cm and 5cm respectively.
$
  TSA = 2\left( {lb + bh + lh} \right) \\
   l = 20cm,b = 10cm,h = 5cm \\
  TSA = 2\left[ {\left( {20 \times 10} \right) + \left( {10 \times 5} \right) + \left( {20 \times 5} \right)} \right] \\
  \to 2\left( {200 + 50 + 100} \right) \\
  \to 2\left( {350} \right) \\
  = 750 \\
  TSA = 750c{m^2} \\
 $
Now, we have to divide the area of the cardboard by the totals surface area of the cuboidal box
$
 n = \dfrac{{Are{a_{cardboard}}}}{{TS{A_{box}}}} \\
  Are{a_{cardboard}} = 14 \times {10^3}c{m^2},TS{A_{box}} = 750c{m^2} \\
  \to n = \dfrac{{14000}}{{750}} \\
  \to n = 186.67 \sim 187 \\
 $
Count cannot be in decimals so we have taken the integer value.
Therefore, 187 cuboidal boxes can be made from the cardboard sheet.

Note: The count is a measure and it has no units. So the same physical quantities must be divided by each other as their units get cancelled and give the count. In the above solution, you should not divide the area of the cardboard by the volume of the cuboidal box; because it results in a wrong answer as area and volume are different and we will not get the count. Be careful while choosing the formula.