Answer
Length = $9.66$ ft
Width = $1.66$ ft
Work Step by Step
First, we can say that the width of the rectangular sheet metal is represented by $x$, and the length can then be expressed as $(x+8)$ ft as it is $8$ ft longer than the width.
So, now we know that:
Length = $(x+8)$ ft
Width = $x$ ft
Area = $16 ft^2$
We also know that the formula for the area of a rectangle is Length $\times$ Width.
Using this formula, we can substitute the values mentioned earlier to form a quadratic equation.
Area = Length $\times$ Width
$16 = (x+8) \times x$
$16 = x^2+8x$ (Subtract 16 from each side to make the equation=0)
$x^2+8x-16=0$
Now that we have our equation, we can solve it using the quadratic formula!
$x=\frac{-b±\sqrt {b^2-4ac}}{2a}$
$x_{1}=\frac{-8+\sqrt {8^2-4(1)(-16)}}{2(1)}$, $x_{2}=\frac{-8-\sqrt {8^2-4(1)(-16)}}{2(1)}$
$x_{1}=\frac{-8+\sqrt {128}}{2}$, $x_{2}=\frac{-8-\sqrt {128}}{2}$
$x_{1}=1.66$, $x_{2}=-9.66$
We want 1 solution only in this case, and it cannot be negative as we are finding the width of a rectangle, which cannot be below 0. So, $x=1.66$.
Length = $(1.66) +8$
=$9.66$ ft
Width = $1.66$ ft
