Question

The perimeter of a rectangle is $ 54 $ inches and its area is $ 182 $ square inches. How do you find the length and width of the rectangle?

Answer 1

Hint: Here we will frame the equations using the given two-word statements and the measures of the perimeter and area and use the formulas for it and by using the substitution method will find out the required measures of length and width.

Complete step-by-step answer:
Let us assume that the length of the rectangle be
Also, assume that the width of the rectangle be
Now, Perimeter of the rectangle is the sum of all the four sides of the rectangle which can be given by –
 $ P = 2(l + b) $
Given that perimeter is $ 54 $ inches, so place it in the above expression.
 $ 54 = 2(l + b) $
The above equation can be re-written as –
 $ 2(l + b) = 54 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ (l + b) = \dfrac{{54}}{2} $
Common factors from the numerator and the denominator cancels each other.
 $ \Rightarrow (l + b) = 27 $ inches …… (A)
Now, Area of the rectangle can be given as –
 $ A = lb $
Given that area is $ 182 $ inches square.
 $ 182 = lb $
Make the required subject “b”
 $ \Rightarrow b = \dfrac{{182}}{l} $ …. (B)
Place the above value in the equation (A)
 $ (l + \dfrac{{182}}{l}) = 27 $
Simplify the above equation –
 $ {l^2} + 182 = 27l $
Move all the terms on one side of the equation –
 $ {l^2} - 27l + 182 = 0 $
Splitting the middle term –
 $ {l^2} - 13l - 14l + 182 = 0 $
Make the pair of first two and last two terms –
 $
  \underline {{l^2} - 13l} - \underline {14l + 182} = 0 \\
  l(l - 13) - 14(l - 13) = 0 \\
  (l - 13)(l - 14) = 0 \;
  $
Now, we have either $ l = 13 $ inches or $ l = 14 $ inches
Now, place the values in equation (A)
When $ l = 13 $ inches, $ b = 14 $ inches
And when $ l = 14 $ inches, $ b = 13 $ inches
So, the correct answer is “$ l = 14 $ inches, $ b = 13 $ inches”.

Note: Always remember when you move any term from one side of the equation to the opposite side then the sign of the term also changes. Area is represented in square units and it is the product of two adjacent sides whereas the circumference is the sum of all the four sides of the rectangle.