Question

If from a wire of length $36$meter a rectangle of the greatest area is made, then its two adjacent sides in meter are

• A. $6,12$
• B. $9,9$
• C. $10,8$
• D. $13,5$

Answer 1

The correct option is B

$9,9$

Explanation for the correct answer:

Step 1: Calculate the area of the rectangle

Let the length of the rectangle be $x.$

Hence, the breadth of the rectangle is $y.$

Therefore,

$$\begin{gathered} \Rightarrow 2(\mathrm{x}+\mathrm{y})=36 \\ \Rightarrow (\mathrm{x}+\mathrm{y})=18 \\ \Rightarrow \mathrm{y}=18-\mathrm{x} \end{gathered}$$

Now, the area of a rectangle is

Area of rectangle $\left(A\right)=\mathrm{Length}\times \mathrm{Breadth}$

$\begin{array}{rcl}\therefore \mathrm{A}& =& \mathrm{xy}\\ & =& \mathrm{x}(18-\mathrm{x})\\ & =& 18\mathrm{x}-{\mathrm{x}}^2\end{array}$

Step 2: Differentiating Area with respect to$x$, we get

$$\begin{gathered} \therefore \frac{d}{dx}\left(A\right)=18x-2{\mathrm{x}}^2 \\ \Rightarrow 0=18-2x\left[\because \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}\right] \\ \Rightarrow 2x=18 \\ \Rightarrow \mathrm{x}=9\mathrm{m} \end{gathered}$$

Step 3: Calculate the value of $y$

$\begin{array}{rcl}\therefore \mathrm{y}& =& 18-\mathrm{x}\\ & =& 18-9\\ & =& 9\mathrm{m}\end{array}$

Therefore, the length of adjacent sides are $9\mathrm{m}\mathrm{and}9\mathrm{m}$.

Hence, the correct answer is an option (B).