Question

the length of a rectangle is 8 m less than three times the width, and the area of the rectangle is 35 m². find the dimensions of the rectangle. length : m width : m

Explanation:

Step1: Define variables

Let the width of the rectangle be $w$ meters. Then the length $l = 3w - 8$ meters.

Step2: Set up the area - formula equation

The area of a rectangle is $A=l\times w$. Given $A = 35$ m², we substitute $l$ and $A$ into the formula: $(3w - 8)\times w=35$.
Expand it to get $3w^{2}-8w - 35 = 0$.

Step3: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 3$, $b=-8$, $c = - 35$), we can use the quadratic formula $w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(-8)^{2}-4\times3\times(-35)=64 + 420 = 484$.
Then $w=\frac{8\pm\sqrt{484}}{2\times3}=\frac{8\pm22}{6}$.
We have two solutions for $w$:
$w_1=\frac{8 + 22}{6}=\frac{30}{6}=5$ and $w_2=\frac{8-22}{6}=\frac{-14}{6}=-\frac{7}{3}$.
Since the width cannot be negative, we take $w = 5$ m.

Step4: Find the length

Substitute $w = 5$ into the length formula $l=3w - 8$. Then $l=3\times5-8=15 - 8 = 7$ m.

Answer:

Length : 7 m
Width : 5 m