Question

isaiah sketches a model of a skateboard ramp. the model has two surfaces on which to skate, represented by sides ab and ad in the diagram. the steepest side of the model, ab, measures 4 inches. what is the length of the other skating surface ad? 2√2 in 2√3 in 4√2 in 4√3 in

Explanation:

Step1: Analyze triangle \(ABC\)

In right - triangle \(ABC\), \(\angle B = 45^{\circ}\), \(\angle ACB=90^{\circ}\), and \(AB = 4\) inches. Using the sine function \(\sin B=\frac{AC}{AB}\). Since \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), we have \(AC = AB\times\sin45^{\circ}\). Substituting \(AB = 4\), we get \(AC=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}\) inches.

Step2: Analyze triangle \(ACD\)

In right - triangle \(ACD\), \(\angle D = 30^{\circ}\), \(\angle ACD = 90^{\circ}\). Using the sine function \(\sin D=\frac{AC}{AD}\). Since \(\sin30^{\circ}=\frac{1}{2}\), then \(AD=\frac{AC}{\sin30^{\circ}}\). Substituting \(AC = 2\sqrt{2}\) and \(\sin30^{\circ}=\frac{1}{2}\), we have \(AD = 4\sqrt{2}\) inches.

Answer:

\(4\sqrt{2}\text{ in}\) (the third option)