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# A rope is made round to cover an area of 154 $c{{m}^{2}}$then the length of the rope is \begin{align} & \left( A \right)66cm \\ & \left( B \right)132cm \\ & \left( C \right)44cm \\ & \left( D \right)88cm \\ \end{align}

Last updated date: 18th Jun 2024
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Hint: To solve the given question, we should know the formulas for the circumference of a circle and the area of a circle. The circumference of a circle is $2\pi r$, and the area of a circle is $\pi {{r}^{2}}$, here r is the radius of the circle. Also, we should know that if a wire is bent into a circular shape, then the circumference of the circle formed equals the length of the wire.

Complete step by step answer:

We are given an area which a wire covers by making it round, we need to find the length of the wire. As we know that if a wire is bent into a circular shape, then the circumference of the circle formed equals the length of the wire. So, we need to find the circumference of the circle formed, to find the length of the wire. Let the radius of the circle is r, using the formula for the area of the circle, we get
$Area=\pi {{r}^{2}}$
Substituting the value of area, we get
$\Rightarrow 154=\dfrac{22}{7}{{r}^{2}}$
Solving the above equation, we get
$\Rightarrow {{r}^{2}}=49$
Taking the square root of both sides, we get
$\Rightarrow r=7cm$
Using the formula for the circumference of the circle, we get
\begin{align} & \Rightarrow \text{circumference}=2\pi r=2\times \dfrac{22}{7}\times 7 \\ & \therefore \text{circumference}=44cm \\ \end{align}
Hence, the length of the wire is $44cm$.

So, the correct answer is “Option C”.

Note: Here, we take the value of $\pi$ to be $\dfrac{22}{7}$. We did this because the value of the area was given as a multiple of 22. For these types of problems, if the value of a given area or circumference is a multiple of 22, then it is better to use the value of $\pi$ as $\dfrac{22}{7}$. If it is not then, take the value of $\pi$ as $3.14$.