Question

The width of a rectangular room is $\dfrac{4}{7}$ of its length, $x$ and its perimeter is $y$. Write an equation connecting $x$and $y$. Find the length of the room when the perimeter is 4400cm.
A. $y = 2x;\,2.2{\text{m}}$
B. $y = \dfrac{{22}}{7}x;\,14{\text{m}}$
C. $y = \dfrac{1}{7}x;\,28{\text{m}}$
D. $y = \dfrac{{11}}{7}x;\,28{\text{m}}$

Answer 1

Hint: Use the formula of perimeter of rectangle; perimeter = 2 (length + breadth). We are given the width of the rectangular room as $\dfrac{4}{7}$ of its length where length is assumed as \[x\] so, using this we will first find the value of width in terms of length. Now, apply the formula of the perimeter of the rectangle and we will let the perimeter be $y$. After this we will substitute the value of width obtained and perimeter as $y$. From this we will get the equation in \[x\] and \[y\]. Since, the perimeter is given as 4400 we will substitute in place of $y$ and find the value of length.

Complete step by step solution:
Width of the rectangular room is $\dfrac{4}{7}$ of its length and perimeter is $y$.
First let the value of length as $x$
Therefore, using the given information, we will find the value of width in terms of length
Thus, we get
$ \Rightarrow w = \dfrac{4}{7}x$ ---equation $\left( 1 \right)$
As we know the formula for perimeter of rectangle, we get,
Perimeter of rectangle $ = 2\left( {l + b} \right)$ ---equation $\left( 2 \right)$
Where $l$ is length, $b$ is breadth.
Next, Let perimeter be $y$and length is $x$ & breadth is $w$
Hence, we get,
$ \Rightarrow y = 2\left( {l + w} \right)$
Now, we will put value of equation $\left( 1 \right)$ in equation $\left( 2 \right)$
We get,
\[
   \Rightarrow y = 2\left( {x + \dfrac{4}{7}x} \right) \\
   \Rightarrow y = 2 \times \dfrac{{11}}{7} \times x \\
   \Rightarrow y = \dfrac{{22}}{7}x - - - - {\text{equation}}\,\left( 3 \right) \\
\]
Thus, we get the equation in terms of \[x\] and \[y\].
As given in the question that perimeter is 4400cm --$\left( 4 \right)$
We will put the value of equation $\left( 4 \right)$in equation $\left( 3 \right)$
Thus, we get,
$
   \Rightarrow 4400 = \dfrac{{22}}{7}x \\
   \Rightarrow 1400 = x = {\text{length}} \\
$
Hence length $ = 1400{\text{ cm}} = 14{\text{ m}}$
Next, substitute the value of length in $w = \dfrac{4}{7}x$ to find the value of width
Thus, we get,
$
   \Rightarrow w = \dfrac{4}{7}\left( {14} \right) \\
   \Rightarrow w = 8 \\
$
Therefore, option B, $y = \dfrac{{22}}{7}x;\,14{\text{m}}$ is correct.

Note: In last give the answer in meters as it is asked in meters in given options. Since the perimeter is given in centimeters but the answer is in meters, use the conversion formula and convert the value from centimeter to meter by using \[1m = 100cm\]. Use the formula of the perimeter of the rectangle \[P = 2\left( {l + b} \right)\].