Question

Given a perimeter of 180, how do you find the length and the width of the rectangle of the maximum area?

Answer 1

Here will first assume the length and width of the rectangle to be any variable. We will then use the formula of the perimeter of the rectangle to find the value of one of the dimensions in terms of variables. Then we will substitute the values of dimensions in terms of variables in the formula for the area of a rectangle. We will then simplify and differentiate the area. We will then equate it with the zero and simplify it to find the length and breadth of the rectangle.

Complete step by step solution:
Here we have been given a perimeter and we have to find the length and the width of the rectangle.
Let the length of the rectangle be $x$ and the width of the rectangle be $y$.
Now, we will find the perimeter of the rectangle using the assumed length and the width of the rectangle and the given perimeter is 180.
We know that the formula of perimeter of the rectangle is given by ${\rm{Perimeter}} = 2\left( {{\rm{Length}} + {\rm{Breadth}}} \right)$
Now, we will substitute the value of perimeter, length, and width of the rectangle here.
$\Rightarrow 180 = 2\left( {x + y} \right)$
Now, we will divide both sides by 2.
$\begin{array}{l} \Rightarrow \dfrac{{180}}{2} = \dfrac{{2\left( {x + y} \right)}}{2}\\\ \Rightarrow 90 = x + y\end{array}$
Subtracting $x$ from both the sides, we get
$\begin{array}{l} \Rightarrow 90 - x = x + y - x\\\ \Rightarrow 90 - x = y\end{array}$
$\Rightarrow y = 90 - x$ …………. $\left( 1 \right)$
Let the area of the rectangle be $A$.
We know that the formula of the area of the rectangle is given by Area of rectangle$=$ Length $\times$Breadth
Now, we will substitute the value of the area, length, and breadth in the formula, we get
$A = x \times y$
Now, we will substitute the value of $y$ from equation $\left( 1 \right)$ in the above equation. Therefore, we get
$\Rightarrow A = x\left( {90 - x} \right)$
Now, multiplying the terms using the distributive property of multiplication, we get
$\Rightarrow A = 90x - {x^2}$
Now, we will differentiate both sides with respect to $x$.
$\Rightarrow \dfrac{{dA}}{{dx}} = \dfrac{{d\left( {90x - {x^2}} \right)}}{{dx}}$
Using the differentiation formula $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$, we get
$\Rightarrow \dfrac{{dA}}{{dx}} = 90 - 2x$
Now, to find the maximum area, $\dfrac{{dA}}{{dx}}$ should be equal to 0. So,
$\begin{array}{l}\dfrac{{dA}}{{dx}} = 0\\\ \Rightarrow 90 - 2x = 0\end{array}$
Subtracting 90 from both sides, we get
$\Rightarrow 90 - 2x - 90 = 0 - 90$
On further simplification, we get
$\Rightarrow - 2x = - 90$
Now, dividing both sides by $- 2$, we get
$\Rightarrow \dfrac{{ - 2x}}{{ - 2}} = \dfrac{{ - 90}}{{ - 2}}$
$\Rightarrow x = 45$
Now, we will substitute the value of $x$ in equation $\left( 1 \right)$. Therefore, we get
$y = 90 - 45$
Subtracting the terms, we get
$\Rightarrow y = 45$

Therefore, the value of length of the rectangle is equal to 45 and breadth of the rectangle is equal to 45.

Note:
We know that a rectangle is defined as a polygon whose opposite sides are parallel to each other and are also equal to each other. Each internal angle of a rectangle is equal to $90^\circ$ and the sum of all interior angles is $360^\circ$. We have used the distributive property of multiplication to simplify the equation. According to the distributive property of multiplication, if $a$, $b$ and $c$ are real numbers then $a \cdot \left( {b + c} \right) = a \cdot b + a \cdot c$.