Question

the \golden rectangle\ is a rectangle that has the following ratio of width to height: $\frac{w}{h}=\frac{h}{w + h}$. if the width 12, what is the height? give answer to the nearest hundredth.

Explanation:

Step1: Substitute \(w = 12\) into the ratio.

Given \(\frac{w}{h}=\frac{h}{w + h}\), substituting \(w = 12\) gives \(\frac{12}{h}=\frac{h}{12 + h}\).

Step2: Cross - multiply.

Cross - multiplying the equation \(\frac{12}{h}=\frac{h}{12 + h}\) results in \(12(12 + h)=h^{2}\).
Expanding the left - hand side: \(144+12h = h^{2}\).

Step3: Rearrange to a quadratic equation.

Rearrange \(144+12h = h^{2}\) to the standard quadratic form \(h^{2}-12h - 144=0\).

Step4: Use the quadratic formula.

The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For \(h^{2}-12h - 144=0\), \(a = 1\), \(b=-12\), and \(c=-144\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-12)^{2}-4\times1\times(-144)=144 + 576=720\).
Then \(h=\frac{12\pm\sqrt{720}}{2}=\frac{12\pm\sqrt{144\times5}}{2}=\frac{12\pm12\sqrt{5}}{2}=6\pm6\sqrt{5}\).
We take the positive root since height cannot be negative. So \(h = 6 + 6\sqrt{5}\approx6+6\times2.2361=6 + 13.4166 = 19.42\).

Answer:

\(19.42\)