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: Find the length of \(AC\) in \(\triangle ABC\)

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

$$AC = 4\times\frac{\sqrt{2}}{2}=2\sqrt{2}$$

Step2: Find the length of \(AD\) in \(\triangle ACD\)

In right - triangle \(ACD\) with \(\angle D = 30^{\circ}\) and \(\angle ACD = 90^{\circ}\), \(\sin D=\frac{AC}{AD}\). Since \(\sin30^{\circ}=\frac{1}{2}\) and \(AC = 2\sqrt{2}\), we can rewrite the formula as \(AD=\frac{AC}{\sin30^{\circ}}\).

$$AD=\frac{2\sqrt{2}}{\frac{1}{2}} = 4\sqrt{2}$$

Answer:

\(4\sqrt{2}\) in.