# One more rectangular-shaped piece of metal siding needs to be cut to cover the exterior of a pole barn. The area of the piece is 30 ft^2. The length is 1 less than 3 times the width. How wide should the metal piece be?

Feb 17, 2016

Draw a diagram to represent the situation.

#### Explanation:

The formula for area of a rectangle is $A = L \times W$.

We know:

L = 3x - 1

W = x

A= 30

All we have to do is solve for x.

$A = L \times W$

$30 = \left(3 x - 1\right) \times x$

$30 = 3 {x}^{2} - x$

$0 = 3 {x}^{2} - x - 30$

This trinomial is factorable. Since it is of the form $y = a {x}^{2} + b x + c$, you must find two numbers that multiply to $a c$ and that add to b. Two numbers that multiply to -90 and that add to -1 are -10 and 9.

$0 = 3 {x}^{2} + 9 x - 10 x - 30$

$0 = 3 x \left(x + 3\right) - 10 \left(x + 3\right)$

$0 = \left(3 x - 10\right) \left(x + 3\right)$

$x = \frac{10}{3} \mathmr{and} - 3$

Since a negative side length is impossible, the width must measure $\frac{10}{4}$. As a result, the width measures $3 \left(\frac{10}{3}\right) - 1 = 9$.

The dimensions of the piece of metal would be $\frac{10}{3} \times 9$ feet.

Practice exercises:

1. A rectangle has an area of $72 c {m}^{2}$. The length measures two more than four times the width. Find the perimeter of the rectangle.

2. A right triangle has two legs and a hypotenuse. The hypotenuse measures $\sqrt{1000}$ inches. The longer leg measures ten less than the double of the shorter leg. Find the area of the triangle.

3. A rectangle has a perimeter of 46 meters. Its area is $126 m e t e r {s}^{2}$.

Good luck!