Question

The perimeter of a rectangle is 36ft, and the area of the rectangle is 72 ft square. How do you find the dimension?

Answer 1

Hint: We know that the area of the rectangle and the perimeter of the rectangle. That is the area of the rectangle is \[A = l \times w\] , the perimeter of the rectangle is \[p = 2(l + w)\] . Using this we need to find the length and width that is the dimensions of the rectangle.

Complete step-by-step answer:
Given,
Perimeter of a rectangle is \[p = 36ft\]
Also given the area of the rectangle is \[A = 72f{t^2}\] .
We know the area of the rectangle is \[A = l \times w\] and the perimeter of the rectangle is \[p = 2(l + w)\] .
Substituting the given data In the formulas we have
 \[72 = l \times w{\text{ }} - - - (1)\]
 \[32 = 2(l + w){\text{ }} - - - (2)\]
From equation (1) we have,
 \[w = \dfrac{{72}}{l}\] , substituting this in equation (2) we have,
 \[32 = 2\left( {l + \dfrac{{72}}{l}} \right)\]
 \[32 = 2l + 2 \times \dfrac{{72}}{l}\]
 \[32 = 2l + \dfrac{{144}}{l}\]
Taking LCM and simplifying we have,
 \[32 = \dfrac{{2{l^2} + 144}}{l}\]
 \[32l = 2{l^2} + 144\]
Rearranging we have
 \[2{l^2} - 32l + 144 = 0\]
Divide the whole equation by 2, we have
 \[{l^2} - 18l + 72 = 0\]
We can split the middle term using factorization method we have,
 \[{l^2} - 12l - 6l + 72 = 0\]
 \[l(l - 12) - 6(l - 12) = 0\]
Taking \[(l - 12)\] common we have,
 \[(l - 12)(l - 6) = 0\]
Using zero product principle we have,
 \[(l - 12) = 0\] and \[(l - 6) = 0\]
 \[ \Rightarrow l = 12\] and \[l = 6\]
Since the length can be width and width can be length, the sides of rectangle measure are 12 and 6.
Hence the dimensions are 12 ft and 6 ft
So, the correct answer is “12 ft and 6 ft”.

Note: In above we obtained a quadratic equation by simplification. If we are unable to solve the obtained quadratic equation we use quadratic formula or Sirdar’s formula to solve it, that is \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . Here length can be 12ft or 6f and width can be 12ft or 6ft. Here we need the area and the perimeter in the same unit otherwise we convert it to the same unit.