Question

tamika wants to use a sheet of fiberboard 19 inches long to create a skateboard ramp with a \\(20^{\circ}\\) 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:

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<reasoning>

Model the physical scenario

Using the Angle of Elevation and Right Triangle Trigonometry knowledge points
\[

$$\begin{aligned} &\theta = 20^\circ \quad \text{(angle of elevation)}\\ &\text{Hypotenuse } (c) = 19\text{ inches} \quad \text{(length of the ramp)}\\ &\text{Opposite side } (h) = \text{height of the ramp} \end{aligned}$$

\]

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\\ &\sin(20^\circ) = \frac{h}{19} \end{aligned}$$

\]

Solve for the height

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &h = 19 \cdot \sin(20^\circ)\\ &\sin(20^\circ) \approx 0.34202\\ &h \approx 19 \cdot 0.34202 \approx 6.49838\text{ inches} \end{aligned}$$

\]

Round to the nearest tenth

Using the Trigonometric Applications knowledge point
\[
h \approx 6.5\text{ inches}
\]
</reasoning>

<answer>
\(6.5\text{ inches}\)
</answer>

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Answer:

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<reasoning>

Model the physical scenario

Using the Angle of Elevation and Right Triangle Trigonometry knowledge points
\[

$$\begin{aligned} &\theta = 20^\circ \quad \text{(angle of elevation)}\\ &\text{Hypotenuse } (c) = 19\text{ inches} \quad \text{(length of the ramp)}\\ &\text{Opposite side } (h) = \text{height of the ramp} \end{aligned}$$

\]

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\\ &\sin(20^\circ) = \frac{h}{19} \end{aligned}$$

\]

Solve for the height

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &h = 19 \cdot \sin(20^\circ)\\ &\sin(20^\circ) \approx 0.34202\\ &h \approx 19 \cdot 0.34202 \approx 6.49838\text{ inches} \end{aligned}$$

\]

Round to the nearest tenth

Using the Trigonometric Applications knowledge point
\[
h \approx 6.5\text{ inches}
\]
</reasoning>

<answer>
\(6.5\text{ inches}\)
</answer>

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