Chapter 9.8, Problem 5P

To determine

To calculate:

The dimensions of rectangle

Answer to Problem 5P

The dimensions of rectangle are $12ft$ and $5ft$ .

Explanation of Solution

Given information:

  $\begin{array}{l}P=14ft\\ C=25ft\end{array}$

Formula used:

The perimeter of rectangle is given by the following expression:

  $P=2x+2y$

Where,

Length of rectangle: $x$

Width of rectangle: $y$

Calculation:

The perimeter of rectangle is given as:

  $P=2x+2y$

Put the given data:

  $2x+2y=34$

  $x+y=17$  ... (1)

On squaring:

  ${\left(x+y\right)}^2={17}^2$  ... (2)

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

  $C^2=x^2+y^2$  ... (2)

From equation (2):

  $\begin{array}{l}{\left(x+y\right)}^2={17}^2\\ x^2+y^2+2xy=289\\ {13}^2+2xy=289\\ 2xy=120\\ xy=60\\ y=\frac{60}{x}\end{array}$

Now, put value $y=\frac{60}{x}$ in eq. (1):

  $\begin{array}{l}x+y=17\\ x+\frac{60}{x}=17\\ x^2+60=17x\\ x^2-17x+60=0\\ \left(x-5\right)\left(x-12\right)=0\end{array}$

At $x=12$ :

  $\begin{array}{l}x+y=17\\ 12+y=17\\ y=5\end{array}$

Therefore, length and width of rectangle are calculated as $12ft$ and $12ft$ .