Question

The adjacent sides of a rectangle with a given perimeter is $100\text{cm}$ and enclosing maximum area are.

• A. $10\text{cm and}40\text{cm}$
• B. $20\text{cm and 3}0\text{cm}$
• C. $25\text{cm and}25\text{cm}$
• D. $15\text{cm and}35\text{cm}$

Answer 1

The correct option is C

$25\text{cm and}25\text{cm}$

Explanation for the correct option:

Find the adjacent sides of the rectangle.

Step 1: Given information

Let us assume that,

The adjacent side of rectangle is $x\text{and}y$.

The perimeter of rectangle is $2\left(\text{sum of two adjacent sides length}\right)$

Then the parameter can be expressed as,

$$\begin{gathered} \Rightarrow 2\left(x+y\right)=100\text{cm} \\ \Rightarrow \left(x+y\right)=50\text{cm} \\ \Rightarrow y=\left(50-x\right) \end{gathered}$$

Step 2: Finding $\frac{dA}{dx}$

Since, the area of rectangle is product of two adjacent sides.

The area of the rectangle can be expressed as,

$$\begin{gathered} A=xy \\ A=x\left(50-x\right) \\ A=50x-x^2 \end{gathered}$$

Differentiate the above Equation

$\frac{dA}{dx}=50-2x$

Step 3: Finding the maximum area

For the maximum area can be expressed as,

$\frac{dA}{dx}=0$

Then,

$$\begin{gathered} \Rightarrow 50-2x=0 \\ \Rightarrow -2x=-50 \\ \Rightarrow x=\frac{50}{2} \\ \Rightarrow x=25 \end{gathered}$$

Therefore, getting the value of $y$

$$\begin{gathered} y=50-x \\ y=50-25 \\ y=25 \end{gathered}$$

Hence, option (C) is the correct answer.