Question

rajani wants to use a sheet of fiberboard 31 inches long to create a skateboard ramp with a 19° angle of elevation from the ground. how high will the ramp rise from the ground at its highest end? round your answer to the nearest tenth of an inch if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We know that in a right - triangle (the skateboard ramp forms a right - triangle with the ground), the sine function is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the angle of elevation \(\theta = 19^{\circ}\), the hypotenuse (the length of the fiberboard) \(c = 31\) inches, and we want to find the length of the side opposite the angle \(\theta\) (the height of the ramp) \(h\).
So, \(\sin\theta=\frac{h}{c}\), which can be rewritten as \(h = c\times\sin\theta\).

Step2: Substitute the values

Substitute \(c = 31\) and \(\theta=19^{\circ}\) into the formula. We know that \(\sin(19^{\circ})\approx0.3256\) (using a calculator in degree mode).
Then \(h = 31\times\sin(19^{\circ})\).
\(h=31\times0.3256\).

Step3: Calculate the value

\(h = 31\times0.3256=10.0936\).

Answer:

\(10.1\) inches