Question

part 1 of 4
(a) write an equation in terms of x that represents the given relationship.
the equation is 180 = (2x + 1)(x)(5)
alternate answer:
5x(2x + 1) = 180
part: 1 / 4
part 2 of 4
(b) solve the equation to find the dimensions of the given shape.
the length is ft.

Explanation:

Step1: Simplify the equation

We start with the equation \(5x(2x + 1)=180\). First, divide both sides by 5 to simplify:
\(\frac{5x(2x + 1)}{5}=\frac{180}{5}\)
Which gives \(x(2x + 1)=36\).

Step2: Expand the left - hand side

Expand \(x(2x + 1)\) using the distributive property \(a(b + c)=ab+ac\) (here \(a = x\), \(b = 2x\), \(c = 1\)):
\(2x^{2}+x = 36\)

Step3: Rearrange into standard quadratic form

Subtract 36 from both sides to get the quadratic equation in standard form \(ax^{2}+bx + c = 0\):
\(2x^{2}+x-36 = 0\)

Step4: Solve the quadratic equation

We can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for a quadratic equation \(ax^{2}+bx + c = 0\). Here, \(a = 2\), \(b = 1\), and \(c=- 36\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(1)^{2}-4\times2\times(-36)=1 + 288 = 289\)
Then, \(x=\frac{-1\pm\sqrt{289}}{2\times2}=\frac{-1\pm17}{4}\)
We have two solutions:
\(x_{1}=\frac{-1 + 17}{4}=\frac{16}{4}=4\)
\(x_{2}=\frac{-1-17}{4}=\frac{-18}{4}=-4.5\)
Since \(x\) represents a dimension - related quantity (from the context of the shape's dimensions), it must be positive. So we take \(x = 4\).

Step5: Find the length

The length is given by \(2x + 1\). Substitute \(x = 4\) into \(2x+1\):
\(2\times4+1=8 + 1=9\)

Answer:

The length is \(\boldsymbol{9}\) ft.