Question

Area of a rectangle is 30sq. units and its perimeter is 60 units. What will be the length of the longest side of this rectangle? ​

Answer 1

Answer:

180 is the correct answer

Answer 2

The longest side of the given rectangle is 28.96 units.

Given,

Area of a rectangle = 30 unit²,

Its perimeter = 60 units.

To find,

Length of its longest side.

Solution,

It can be seen here, that the area (let A) and perimeter (let P) of a rectangle are given which are as follows.

A = 30 unit²,

P = 60 units.

Now, for a rectangle having two sides as l and b,

the area is given by

$A = l \times b \hfill ...(1)$

and perimeter is

$P=2(l+b) \hfill ...(2)$

So, for the given rectangle from (1), we get,

$l \times b = 30 \hfill ...(3)$

and, from (2),

$2(l+b) = 60$

$\implies l + b = 30 \hfill ...(4)$

Solving equations (3) and (4), we can determine the sides of the rectangle.

Thus, from (4),

$b = 30 - l$

substituting in (3), we get,

$l(30-l)=30$

$\implies 30l-l^{2} =30$

$l^{2} -30l+30=0 \hfill ...(5)$

The above quadratic equation in (5) can be solved using the quadratic formula, which is

$x = \frac{-b \pm \sqrt{b^{2} -4ac} }{2a}$

note that 'b' in the above formula is not the side of the rectangle.

Here, from quadratic equation in (5), we have,

a = 1, b = -30, and c = 30.

So,

$l = \frac{-(-30) \pm \sqrt{(-30)^{2} -4(1)(30)} }{2(1)}$

$\implies l = \frac{30 \pm \sqrt{900 -120} }{2}$

$\implies l = \frac{30 \pm \sqrt{780} }{2}$

$\implies l =15 \pm \sqrt{195}$

⇒ l = 15 ± 13.96

⇒ l = 28.96, 1.04.

Now, from (4),

if l = 28.96,

b = 30 - 28.96

⇒ b = 1.04.

and, if l = 1.04,

b = 30 - 1.04

⇒ b = 28.96.

So, the possible pairs of the sides of the given rectangle are

(l, b) = (28.96, 1.04), or

(l, b) = (1.04, 28.96).

However, in both cases, the longest side = 28.96 units.

Therefore, the longest side of the given rectangle is 28.96 units.

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