isaiah sketches a model of a skateboard ramp. the model has two surfaces on which to skate, represented by…

Answer:

$4\sqrt{2}\text{ in.}$

Explanation:

Step1: Analyze right - triangle ABC

In right - triangle $ABC$, $\angle B = 45^{\circ}$ and $\angle ACB=90^{\circ}$, and $AB = 4$ inches. Since $\angle B = 45^{\circ}$, triangle $ABC$ is a 45 - 45 - 90 triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. Let $AC = BC=x$. Using the Pythagorean theorem $AB^{2}=AC^{2}+BC^{2}$, and since $AC = BC$, we have $AB^{2}=2x^{2}$. Given $AB = 4$, then $4^{2}=2x^{2}$, $16 = 2x^{2}$, $x^{2}=8$, $x = AC=2\sqrt{2}$ inches.

Step2: Analyze right - triangle ACD

In right - triangle $ACD$, $\angle D = 30^{\circ}$, $\angle ACD = 90^{\circ}$, and we know the side opposite the $30^{\circ}$ angle is $AC = 2\sqrt{2}$ inches. In a 30 - 60 - 90 triangle, the ratio of the sides is $1:\sqrt{3}:2$. If the side opposite the $30^{\circ}$ angle is $a$, the hypotenuse $AD$ (the side we want to find) is related to the side opposite the $30^{\circ}$ angle by the formula $AD = 2AC$. Since $AC = 2\sqrt{2}$ inches, then $AD=4\sqrt{2}$ inches.