Question

The area of a rectangle is 192 square cm. and its perimeter is 56 cm. Find the length.

Answer 1

Hint—The area of the rectangle and perimeter of the rectangle is given. So, use the formula of both and form the equations and solve them using substitution method and find the value of length. The formula for area of rectangle is given by \[A = l \times b\] and formula for perimeter of rectangle is given by \[P = 2\left( {l + b} \right)\] .While solving the problem using substitution method, use the middle term splitting method to find the values of \[l\]. We can use the elimination method also to evaluate the value of length.

Complete step-by-step solution
Consider the given information and let the length as \[l\] and breadth as \[b\] of the rectangle.
The area of the rectangle is given as 192 sq. cm.
We will write the formula of the area of the rectangle and put it equal to 192.
We get,
\[l \times b = 192\] ---(1)
The perimeter of the rectangle is given as 56 cm.
We will write the formula of the perimeter of the rectangle and put it equal to 56.
We get,
\[2\left( {l + b} \right) = 56\] ---(2)
Simplify the equation (2) and find the value of breadth in terms of length.
Thus, we get,
\[
   \Rightarrow \left( {l + b} \right) = \dfrac{{56}}{2} \\
   \Rightarrow b = 28 - l \\
\]
Now, substitute the value of breadth in equation (1),
\[
  l \times \left( {28 - l} \right) = 192 \\
  {l^2} - 28l + 192 = 0 \\
 \]
We will use the middle-term splitting method to find the roots of the obtained equation,
\[
   \Rightarrow {l^2} - 16l - 12l + 192 = 0 \\
   \Rightarrow l\left( {l - 16} \right) - 12\left( {l - 16} \right) = 0 \\
   \Rightarrow \left( {l - 12} \right)\left( {l - 16} \right) = 0 \\
\]
Now, we will use the zero-factor property,
\[ \Rightarrow \left( {l - 12} \right) = 0\] or \[l - 16 = 0\]
\[ \Rightarrow l = 12\] or \[l = 16\]
Thus, we get the 2 values of \[l = 12,16\].

Note: We have used the substitution method to find the value of length, we can also use other methods to find the value. We have got 2 values of length, as we substitute it in equation (1) we will get, \[12 \times b = 192\] or \[16 \times b = 192\] this shows that the value of breadth is \[12\] or 16 when value of length is 16 or 12 respectively.