Question

the length of a rectangular field is 3 root 5 + 3 root 2 find the measure of its breadth such that the area of rectangle is a rational number

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Answer 1

The breadth of the rectangle is $3\sqrt{5}-3\sqrt{2}$

Step-by-step explanation:

Let the breadth of the rectangle be 'B'.

Given:

The length of the rectangle is, $L=3\sqrt{5}+3\sqrt{2}$

The length is an irrational number. The area of a rectangle is given as the product of length and its breadth.

So, in order to get the area as rational number, the product of the length and breadth should be rational. Now, the length is irrational number. So the breadth should also be irrational and conjugate of the length because the product of an irrational and its conjugate is always a rational number.

The conjugate of the length is $B=3\sqrt{5}-3\sqrt{2}$

Therefore, the measure of its breadth such that the area of rectangle is a rational number is $3\sqrt{5}-3\sqrt{2}$.

$Area=Length\times Breadth$

$Area=(3\sqrt{5}+3\sqrt{2})(3\sqrt{5}-3\sqrt{2})$

$Area=(3\sqrt{5})^2-(3\sqrt{2})^2$

$Area=9(5)-9(2)=45-18=27$

Therefore, the area is a rational number. So, the breadth chosen is correct.