Name: What is an Equation?
Uploaded: 2019-07-19
Duration: 3 min 1 s
Description: If I have ten oranges and you have x number of oranges which add up to 30 oranges, then we have an equation ready, which is 10 + x = 30. It is that simple! L...

Question

A

10

metre long rope is to be cut into two pieces and a square is to be made using each. The difference in the areas enclosed must be

1 \frac{1}{4}

square metres. How should it be cut?

Accepted Answer

Let the $10 m$ long rope be cut into two pieces of the lengths $x m$ (bigger piece) and $y m$ (smaller piece).

So, we have $x + y = 10$ -----(i)

Given that the two pieces and a square is to be made using each piece and the difference in the areas enclosed is $1 \frac{1}{4}$ square metres.

Hence, we have $x^{2} - y^{2} = 1 \frac{1}{4}$

$\Rightarrow x^{2} - y^{2} = \frac{5}{4}$

$\Rightarrow (x + y) (x - y) = \frac{5}{4}$ [Use $a^{2} - b^{2} = (a + b) (a - b)$ ]

$\Rightarrow 10 \times (x - y) = \frac{5}{4}$ [Substitute the value of $x + y$ from (i)]

$\Rightarrow (x - y) = \frac{5}{4} \times \frac{1}{10} = \frac{1}{8}$

$\Rightarrow x - y = \frac{1}{8}$ -----(ii)

Adding (i) and (ii), we get, $x + y + x - y = 10 + \frac{1}{8}$

$\Rightarrow 2 x = \frac{80 + 1}{8}$

$\Rightarrow x = \frac{81}{2 \times 8} = \frac{81}{16} = 5 \frac{1}{16} m$ .

From (i), we get, $x + y = 10$

$\Rightarrow y = 10 - x = 10 - \frac{81}{16} = \frac{160 - 81}{16} = \frac{79}{16} = 4 \frac{15}{16} m$ .

Therefore, the $10 m$ piece is to be cut in pieces whose lengths are $5 \frac{1}{16} m$ and $4 \frac{15}{16} m$ , respectively.

Kerala Board Solutions for Chapter: Pairs of Equations, Exercise 3: Exercise 3

Kerala Board Mathematics Solutions for Exercise - Kerala Board Solutions for Chapter: Pairs of Equations, Exercise 3: Exercise 3

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