Question

sergei buys a rectangular rug for his living room. he measures the diagonal of the rug to be 18 feet. the length of the rug is 3 feet longer than the width. what are the approximate dimensions of the rug? round each dimension to the nearest tenth of a foot. 10.4 feet by 7.4 feet 11.1 feet by 8.1 feet 13.4 feet by 10.4 feet 14.1 feet by 11.1 feet

Explanation:

Step1: Let the width be $x$ feet.

The length is $x + 3$ feet. Using the Pythagorean theorem ($a^{2}+b^{2}=c^{2}$, where $c$ is the diagonal, $a$ is the width and $b$ is the length), we have $x^{2}+(x + 3)^{2}=18^{2}$.

Step2: Expand the equation.

$x^{2}+x^{2}+6x + 9=324$. Combine like - terms to get $2x^{2}+6x+9 - 324 = 0$, so $2x^{2}+6x-315 = 0$.

Step3: Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.

For the equation $2x^{2}+6x - 315=0$, $a = 2$, $b = 6$, and $c=-315$. First, calculate the discriminant $\Delta=b^{2}-4ac=(6)^{2}-4\times2\times(-315)=36 + 2520=2556$. Then $x=\frac{-6\pm\sqrt{2556}}{4}=\frac{-6\pm50.56}{4}$.

Step4: Find the positive value of $x$.

We take the positive root $x=\frac{-6 + 50.56}{4}=\frac{44.56}{4}=11.14\approx11.1$ (we ignore the negative root since width cannot be negative).

Step5: Find the length.

The length is $x + 3=11.1+3 = 14.1$ feet.

Answer:

14.1 feet by 11.1 feet