Question

The areas of a square-shaped tab and rectangle-shaped tab are equal. The perimeter of the rectangular tab is 40 cm. What is its area?

Answer 1

Hint: Here, we will first take the perimeter of the rectangular tab, \[2\left( {l + b} \right)\], where \[l\] is the length and \[b\] is the breadth is 40 cm. Then we will take only integers value as length of side and form a different set and after that we will get the combination that gives the same area as above we show in different length and breadth. Then we will find the area of the square to be found by giving different integers values and when both the value match is our required area.

Complete step-by-step answer:
We are given that the areas of a square-shaped tab and rectangle-shaped tab are equal.

We know that the formula of area of square is \[{a^2}\], where \[a\] is the side of a square and the area of rectangle \[lb\], where \[l\] is the length and \[b\] is the breadth.
We are also given that the perimeter of the rectangular tab, \[2\left( {l + b} \right)\], where \[l\] is the length and \[b\] is the breadth is 40 cm, we get
\[ \Rightarrow 2\left( {l + b} \right) = 40\]
Dividing the above equation by 2 on both sides, we get
\[
   \Rightarrow \dfrac{{2\left( {l + b} \right)}}{2} = \dfrac{{40}}{2} \\
   \Rightarrow l + b = 20 \\
 \]
Now we will take only integer values as the length of the side and form a different set.
If length is 19 cm and width is 1 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 19 \times 1 \\
   \Rightarrow 19{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 18 cm and width is 2 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 18 \times 2 \\
   \Rightarrow 36{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 17 cm and width is 3 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 17 \times 3 \\
   \Rightarrow 51{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 18 cm and width is 4 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 18 \times 4 \\
   \Rightarrow 72{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 16 cm and width is 5 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 16 \times 5 \\
   \Rightarrow 80{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 15 cm and width is 6 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 15 \times 6 \\
   \Rightarrow 90{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 14 cm and width is 7 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 14 \times 7 \\
   \Rightarrow 98{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 13 cm and width is 8 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 13 \times 8 \\
   \Rightarrow 104{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 12 cm and width is 9 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 12 \times 9 \\
   \Rightarrow 108{\text{ c}}{{\text{m}}^2} \\
 \]
If length is 11 cm and width is 10 cm, then we will find the area of rectangle, we get
\[
   \Rightarrow 11 \times 10 \\
   \Rightarrow 110{\text{ c}}{{\text{m}}^2} \\
 \]
After that we will get the combination that gives the same area as above we show in different length and breadth.
Now we will find the area of square to be found by giving different integers values to \[a\], we get
\[ \Rightarrow {1^2} = 1\]
\[ \Rightarrow {2^2} = 2\]
\[ \Rightarrow {3^2} = 9\]
\[ \Rightarrow {4^2} = 16\]
\[ \Rightarrow {5^2} = 25\]
\[ \Rightarrow {6^2} = 36\]
Since the area of the square and a rectangle are equal, we can see that in both areas, we can have only one common area, that is, 36 cm\[^2\].
So, the area of the rectangle is 36 cm\[^2\].


Note: This problem is lengthy as we have to start by assuming the values of length and breadth such that they are equal to the number. We should know the formulae of area of square is \[{a^2}\], where \[a\] is the side of a square and the area of rectangle \[lb\], where \[l\] is the length and \[b\] is the breadth. Do not forget to write the units with the final answer.