Question

The diagonal of a rectangular rug is √II7 ft. The area of the rug is 54 ft². Find the length and width of the rug.

Answer 1

The area of a rectangle is computed as follows:

A = length*height

In this case, the area is 54 ft², then:

54 = xy

Applying the Pythagorean theorem with the height, length, and diagonal of the rectangle we get:

`\begin{gathered} c^2=a^2+b^2 \\ \sqrt[]{117}^2=x^2+y^2 \\ 117^{}=x^2+y^2 \end{gathered}`

Isolating x from the first equation:

54/y = x

Substituting this result into the second equation:

`\begin{gathered} 117=((54)/(y))^2+y^2 \\ 117=(2916)/(y^2)^{}+y^2 \\ \text{Multiplying at both sides by y}^2 \\ 117y^2=(2916)/(y^2)y^2+y^2y^2 \\ 0=y^4-117y^2+2916 \end{gathered}`

Replacing

`\begin{gathered} t=y^2 \\ t^2=(y^2)^2=y^4 \\ 0=t^2-117t+2916 \end{gathered}`

Applying the quadratic formula:

`\begin{gathered} t_(1,2)=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ t_(1,2)=\frac{117\pm\sqrt[]{(-117)^2-4\cdot1\cdot2916}}{2\cdot1} \\ t_(1,2)=\frac{117\pm\sqrt[]{2025}}{2} \\ t_1=(117+45)/(2)=81 \\ t_2=(117-45)/(2)=36 \end{gathered}`

Therefore, in terms of the original variable, y, the solutions are:

`\begin{gathered} y^2_{}=t \\ y=\sqrt[]{t_1}=\sqrt[]{81}=\pm9 \\ y=\sqrt[]{t_2}=\sqrt[]{36}=\pm6 \end{gathered}`

The negative results have no sense in this case, then the length and width are 9 ft and 6 ft, or vice versa.