Chapter 9.8, Problem 8P

To determine

To calculate:

The dimensions of rectangle

Answer to Problem 8P

The dimensions of rectangle are $\sqrt{5}$ and $2\sqrt{5}$ .

Explanation of Solution

Given information:

  $\begin{array}{l}A=10\\ C=5\end{array}$

Formula used:

The area of rectangle is given by the following expression:

  $A=xy$

Where,

Length of rectangle: $x$

Width of rectangle: $y$

Calculation:

The area of rectangle is given as:

  $A=xy$

Put the given data:

  $10=xy$

  $y=\frac{10}{x}$  ... (1)

The diagonal length of the rectangle is hypotenuse which is given by the following expression:

  $C^2=x^2+y^2$  ... (2)

  $\begin{array}{l}{5}^2=x^2+y^2\\ x^2+{\left(\frac{10}{x}\right)}^2=25\\ x^2+\frac{100}{x^2}=25\\ x^4-25x^2+100=0\\ \left(x^2-5\right)\left(x^2-20\right)=0\\ x^2=5OR\text{}{x}^2=20\\ x=\sqrt{5}x=2\sqrt{5}\end{array}$

Now, put value $x=\sqrt{5}$ in eq. (1):

  $\begin{array}{l}y=\frac{10}{x}\\ y=\frac{10}{\sqrt{5}}\\ y=2\sqrt{5}\end{array}$

Therefore, length and width of rectangle are calculated as $\sqrt{5}$ and $2\sqrt{5}$ .