Question

part 1 of 3
(a) write an equation in terms of x that represents the given relationship.
the equation is $216 = \frac{1}{2}x(x + 6)$.
part: 1 / 3
part 2 of 3
(b) solve the equation to find the dimensions of the given shape. be sure to include t
the base is $\square$.
$\text{yd}$ $\text{yd}^2$ $\text{yd}^3$

Explanation:

Step1: Multiply both sides by 2

To eliminate the fraction, we multiply both sides of the equation \(216 = \frac{1}{2}x(x + 6)\) by 2.
\(2\times216 = 2\times\frac{1}{2}x(x + 6)\)
\(432=x(x + 6)\)

Step2: Expand the right - hand side

Expand \(x(x + 6)\) using the distributive property \(a(b + c)=ab+ac\) (here \(a = x\), \(b=x\), \(c = 6\)).
\(x(x + 6)=x^{2}+6x\)
So our equation becomes \(x^{2}+6x - 432=0\)

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 6\), \(c=- 432\)), we can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor the equation. Let's try factoring.
We need two numbers that multiply to \(ac=1\times(-432)=-432\) and add up to \(b = 6\).
The numbers are 24 and - 18 since \(24\times(-18)=-432\) and \(24+( - 18)=6\).
So we can factor the quadratic equation as:
\(x^{2}+24x-18x - 432 = 0\)
\(x(x + 24)-18(x + 24)=0\)
\((x + 24)(x - 18)=0\)

Step4: Find the roots

Set each factor equal to zero:
\(x+24 = 0\) or \(x - 18=0\)
\(x=-24\) or \(x = 18\)
Since the dimension cannot be negative, we discard \(x=-24\). So \(x = 18\) (the base, as the unit is yards (yd)).

Answer:

The base is \(\boldsymbol{18}\) yd.