Question

Find the length of the shortest side of a rectangle which has an area of 32 square feet and a perimeter of 24 feet. (in feet)
(a) 1
(b) 2
(c) 3
(d) 4
(e) 8

Answer 1

Hint: To solve the given question, we will first find out what a rectangle is. Then, we will make use of the fact that the perimeter of a rectangle is given by Perimeter = 2(l + b). We will assume that the length of the rectangle is x and the breadth of the rectangle is y. By applying the formula, we will get a relation between x and y. Then, we will use the formula, \[\text{Area}=l\times b\] to obtain another relation in x and y. From these relations, we will derive a quadratic equation in x, which we will solve with the help of factorization.

Complete step-by-step answer:
Before we start to solve the question given, we will first find out what a rectangle is. A rectangle is a polygon having 4 sides in which adjacent sides are perpendicular to each other and the opposite pair of sides are equal. Let us assume that the length of the rectangle is x and the breadth of the rectangle is y. Now, it is given in the question that the perimeter of the given rectangle is 24 feet. We know that, if l is the length and b is the breadth of a rectangle then, its perimeter is given by the formula
\[\text{Perimeter}=2\left( l+b \right)\]
Thus, we will get,
\[24=2\left( x+y \right)\]
\[\Rightarrow x+y=12\]
\[\Rightarrow y=12-x......\left( i \right)\]
Now, it is given in the question that the area of the given rectangle is 32 square feet. We know that the area of the square is given by the formula
\[\text{Area}=\text{length}\times \text{breadth}\]
\[\Rightarrow 32=xy....\left( ii \right)\]
Now, we will put the value of y from (i) to (ii). Thus, we will get,
\[\Rightarrow 32=x\left( 12-x \right)\]
\[\Rightarrow x\left( 12-x \right)=32\]
\[\Rightarrow 12x-{{x}^{2}}=32\]
\[\Rightarrow {{x}^{2}}-12x+32=0\]
Now, we will solve the above quadratic equation by the method of factorization. Thus, we have,
\[\Rightarrow {{x}^{2}}-8x-4x+32=0\]
\[\Rightarrow x\left( x-8 \right)-4\left( x-8 \right)=0\]
\[\Rightarrow \left( x-8 \right)\left( x-4 \right)=0\]
\[\Rightarrow x=8;x=4\]
Here, x = 4 is the smallest root so the length of the shortest side will be 4 feet.
Hence, option (d) is the right answer.

Note: We can also solve the quadratic equation obtained by the help of the quadratic formula. We know that if the quadratic equation is of the form \[a{{x}^{2}}+bx+c=0,\] then its roots are given by
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
In our case, the quadratic equation is \[{{x}^{2}}-12x+32.\] Thus, we will get,
\[x=\dfrac{-\left( -12 \right)\pm \sqrt{{{\left( -12 \right)}^{2}}-4\left( 1 \right)\left( 32 \right)}}{2\left( 1 \right)}\]
\[\Rightarrow x=\dfrac{12\pm \sqrt{144-128}}{2}\]
\[\Rightarrow x=\dfrac{12\pm \sqrt{16}}{2}\]
\[\Rightarrow x=\dfrac{12\pm 4}{2}\]
\[\Rightarrow x=\dfrac{12+4}{2};x=\dfrac{12-4}{2}\]
\[\Rightarrow x=\dfrac{16}{2};x=\dfrac{8}{2}\]
\[\Rightarrow x=8;x=4\]
Thus the smallest side will have the length 4 feet.