Question

Find the dimensions of a rectangle whose perimeter is \[28meters\] and whose area is \[40\text{square meters}\]?

Answer 1

Hint: We are asked to find the dimension of the rectangle, which means that we are supposed to find the length and breadth of the rectangle. Firstly, we will be finding the length from the perimeter and then we will be substituting it in the formula of the area of the rectangle and equate it to \[40\]. Upon solving the obtained equation, we obtain the required length and breadth of the triangle.

Complete step by step answer:
 Now let us have a brief regarding the properties of a rectangle. We know that a rectangle is a quadrilateral whose opposite sides are parallel and are equal. Each of the interior is a right angle which means that its measure is \[{{90}^{\circ }}\]. The diagonals of the rectangle bisect each other. Also the rectangles have the diagonals of equal length. The sum of interior angles of a rectangle is \[{{360}^{\circ }}\].
Now let us find the dimensions of the rectangle whose perimeter is \[28meters\] and whose area is \[40\text{square meters}\].
Let us consider the perimeter of the rectangle and extract the length from it.
\[\begin{align}
  & 2\left( l+b \right)=28 \\
 & \left( l+b \right)=14 \\
 & l=14-b \\
\end{align}\]
We get the value of the length as \[l=14-b\].
Now let us consider the area of the rectangle i.e. \[a=l\times b\].
Since we know the value of area and length, upon substituting we get,
\[\begin{align}
  & \Rightarrow 14b-{{b}^{2}}=40 \\
 & \Rightarrow -{{b}^{2}}+14b-40=0 \\
 & \Rightarrow {{b}^{2}}-14b+40=0 \\
\end{align}\]
On further solving by the factorization method, we obtain the values,
\[\begin{align}
  & \Rightarrow {{b}^{2}}-14b+40=0 \\
 & \Rightarrow {{b}^{2}}-4b-10b+40=0 \\
 & \Rightarrow b\left( b-4 \right)-10\left( b-4 \right)=0 \\
 & \Rightarrow \left( b-4 \right)\left( b-10 \right)=0 \\
 & \Rightarrow b=4 \text{or} 10 \\
\end{align}\]
We obtain the value of \[b\] as \[4\ or\ 10\] \[meters\]
\[\therefore \] If the value of breadth is \[4\] then the value of length is \[10\] and vice versa.
The rectangle is shown below:

Note: We must always denote the dimensions of the figure with its units if mentioned. This is a common error that would be committed. The above solved problem can also be solved by finding out the perimeter first and then finding the length. Anyhow, we obtain the same answers in both the cases.