Question

1. carmen is building a ramp for her remote - control car. the cross - section of the ramp makes a 30° - 60° - 90° right triangle, with the longer leg along the ground. if carmen wants the ramp to be 8 feet long. exactly how many feet long must the base and height of the ramp be? base: height:

Explanation:

Step1: Recall the properties of a 30 - 60 - 90 triangle

In a 30 - 60 - 90 right triangle, if the hypotenuse is \(c\), the shorter leg (opposite the 30° angle) is \(a=\frac{c}{2}\), and the longer leg (opposite the 60° angle) is \(b = \frac{\sqrt{3}}{2}c\). Here, the hypotenuse \(c = 8\) feet (the length of the ramp).

Step2: Calculate the height (shorter leg)

The height \(h\) (shorter leg) of the 30 - 60 - 90 triangle: \(h=\frac{c}{2}\). Substitute \(c = 8\) into the formula, we get \(h=\frac{8}{2}=4\) feet.

Step3: Calculate the base (longer leg)

The base \(b\) (longer leg) of the 30 - 60 - 90 triangle: \(b=\frac{\sqrt{3}}{2}c\). Substitute \(c = 8\) into the formula, we get \(b=\frac{\sqrt{3}}{2}\times8 = 4\sqrt{3}\) feet.

Answer:

Base: \(4\sqrt{3}\) feet, Height: \(4\) feet