Question

Of all rectangles with a perimeter of $14$ which one has the maximum area? (Give the dimensions.) Let $A$ be the area of the rectangle. What is the objective function in terms of the width of the rectangle, $w$?

Answer 1

Finding the width of the rectangle, $w$:

Given perimeter of the rectangle is $14$

We know the equation for perimeter, $P=2(l+w)$, where $l$ is length of the rectangle and $w$ is width of the rectangle.

$$\begin{gathered} \therefore P=2(l+w)\left[\because P=14\right] \\ \Rightarrow 14=2(l+w) \\ \Rightarrow 7=l+w \\ \Rightarrow l=7-w---------\left(1\right) \end{gathered}$$

Area of a rectangle,

$A=l\times w$

Substituting $\left(1\right)$ in the above equation,

$A=\left(7-w\right)\times w$

We have to substitute values from $0$ to $7$ for $w$ and find the largest value for $A$

Here, we get the largest value when length and width are same.

That is, $l=w=\frac{7}{2}$

Hence,square of given permeter will have maximum area and the function in terms of width of the rectangle is $A=\left(7-w\right)\times w$.