Question

A rope of length 40 meters is cut into pieces and two squares are made on the floor with them. The sum of the areas of the enclosed is 58 square meters.
(a) If the length of one piece is taken as x, what is the length of the other piece?
(b) What are the lengths of the sides of the squares?
(c) Write the given fact about the area as an algebraic equation.
(d) What is the length of each piece?

Answer 1

Hint: In this question, we will add areas of both the squares and equate it with the total area given to find the value of x. Also, write the length of two pieces in variable x only.

Complete step-by-step solution -
The length of the rope = 40 cm
The length of one piece = x
Therefore, the length of the other piece = 40 – x
The length of the sides of the squares is given by dividing the total length of the rope by 4, as a square has 4 sides.
Therefore, the length of the sides \[=\dfrac{x}{4}\text{ and }\dfrac{40-x}{4}\]
The sum of the areas of both the squares \[=58c{{m}^{2}}\]
\[\Rightarrow {{\left( \dfrac{x}{4} \right)}^{2}}+\left( \dfrac{40-x}{4} \right)=58\]
\[\Rightarrow {{\dfrac{x}{16}}^{2}}+\dfrac{1600-80x+{{x}^{2}}}{16}=58\]
\[\Rightarrow 2{{x}^{2}}-80x+1600=58\times 16\]
Dividing throughout by 2, we get,
\[\Rightarrow {{x}^{2}}-40x+800=464\]
\[\Rightarrow {{x}^{2}}-40x+800-464=0\]
\[\Rightarrow {{x}^{2}}-40x+336=0\]
Therefore, the length of each piece,
\[\Rightarrow {{x}^{2}}-40x+336=0\]
\[\Rightarrow \left( x-28 \right)\left( x-12 \right)=0\]
\[\Rightarrow x-28=0;x-12=0\]
\[\Rightarrow x=28;x=12\]
Therefore, the length of each piece is 28m and 12 m respectively.

Note: Remember that the value of x is the sum of all sides of the square and not just a single side. To find a single side, we divide the sum by 4. For solving quadratic equations we can also use the quadratic formula.