Question

question 1 of 4, step 1 of 1 a rectangle has a length of 15 yards less than 2 times its width. if the area of the rectangle is 638 square yards, find the length of the rectangle. answer how to enter your answer (opens in new window) yards keypad keyboard shortcuts

Explanation:

Step1: Let the width be $w$ yards.

The length $l = 2w - 15$ yards.

Step2: Use the area formula $A=l\times w$.

We know $A = 638$, so $(2w - 15)w=638$.
Expanding gives $2w^{2}-15w - 638 = 0$.

Step3: Solve the quadratic equation $2w^{2}-15w - 638 = 0$.

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 2$, $b=-15$, $c = - 638$), the quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(-15)^{2}-4\times2\times(-638)=225 + 5104=5329$.
Then $w=\frac{15\pm\sqrt{5329}}{4}=\frac{15\pm73}{4}$.
We get two solutions for $w$: $w_1=\frac{15 + 73}{4}=\frac{88}{4}=22$ and $w_2=\frac{15-73}{4}=\frac{-58}{4}=-14.5$. Since the width cannot be negative, $w = 22$ yards.

Step4: Find the length.

Substitute $w = 22$ into the length formula $l=2w - 15$.
$l=2\times22-15=44 - 15=29$ yards.

Answer:

29 yards