Question

the length of a rectangle is 11 m less than three times the width, and the area of the rectangle is 70 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 $l = 3w - 11$ meters.

Step2: Set up area - formula equation

The area of a rectangle is $A=l\times w$. Given $A = 70$ m², we substitute $l$ and $A$ into the formula: $(3w - 11)\times w=70$.

Step3: Expand the equation

Expand $(3w - 11)w$ to get $3w^{2}-11w = 70$. Rearrange it to the standard quadratic - form $3w^{2}-11w - 70 = 0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 3$, $b=-11$, $c = - 70$), we can use the quadratic formula $w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor. Factoring $3w^{2}-11w - 70=(3w + 14)(w - 5)=0$.
Setting each factor equal to zero gives $3w+14 = 0$ or $w - 5=0$.
From $3w+14 = 0$, we get $w=-\frac{14}{3}$, but the width cannot be negative.
From $w - 5=0$, we get $w = 5$ m.

Step5: Find the length

Substitute $w = 5$ into the length formula $l=3w - 11$. Then $l=3\times5-11=15 - 11 = 4$ m.

Answer:

Length: 14 m
Width: 5 m