Chapter 1.1, Problem 53E

To determine

To find: The function of the area of the rectangle in terms of the length of one of its sides and state its domain.

Answer to Problem 53E

The formula for the function that represents the area of the rectangle in terms of its length is $A\left(l\right)=10l-l^2$.

The domain of the function of the area (A) of the rectangle in terms of its length is $0<l<10$.

In case the length is to be larger than the breadth, the domain of A is $5<l<10$.

Explanation of Solution

Given:

The perimeter of the rectangle is 20 m.

Formula used:

Area of the rectangle, $A=l\times b$, where ‘l’ is the length of the rectangle and ‘b’ is the breadth of the rectangle.

Perimeter of the rectangle, $P=2\left(l+b\right)$, where ‘l’ is the length of the rectangle and ‘b’ is the breadth of the rectangle.

Calculation:

Let the length of the rectangle be l and the breadth be b.

Since the perimeter of the rectangle is 20m,

$2\left(l+b\right)=20\text{m}$ (1)

Express equation (1) in terms of length.

$\begin{array}{l}2l+2b=20\text{m}\\ 2b=20-2l\\ b=\frac{20-2l}{2}\\ b=10-l\end{array}$

Therefore, the function that represents the area of the rectangle in terms of l is $A\left(l\right)=l\left(10-l\right)$.

That is, $A\left(l\right)=10l-l^2$.

The domain of A is $0<l<10$ as the breadth b cannot take the negative values. In the case of l being larger than b, the domain of A is $5<l<10$.