Question

katherine wants to use a sheet of fiberboard 18 inches long to create a skateboard ramp with a \\(17^{\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>

Identify given values and target unknown

We are given a right triangle representing a skateboard ramp:

• The length of the ramp (hypotenuse) is \(18\text{ inches}\).
• The angle of elevation from the ground is \(17^\circ\).
• The target unknown is the height of the ramp (opposite side), which we will denote as \(h\).

Apply right triangle trigonometry

Using the Right Triangle Trigonometry concept, we relate the opposite side and the hypotenuse using the sine function:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]
Using the Angle of Elevation of \(17^\circ\):
\[
\sin(17^\circ) = \frac{h}{18}
\]

Solve for the height

Isolating the variable \(h\):
\[
h = 18 \cdot \sin(17^\circ)
\]

Calculate the numerical value

Using a calculator to find the value of \(\sin(17^\circ)\):
\[
\sin(17^\circ) \approx 0.29237
\]
Multiply by \(18\):
\[
h \approx 18 \cdot 0.29237 \approx 5.2627\text{ inches}
\]

Round to the nearest tenth

Rounding \(5.2627\) to the nearest tenth of an inch:
\[
h \approx 5.3\text{ inches}
\]
</reasoning>

<answer>
\(5.3\)
</answer>

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Answer:

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<reasoning>

Identify given values and target unknown

We are given a right triangle representing a skateboard ramp:

• The length of the ramp (hypotenuse) is \(18\text{ inches}\).
• The angle of elevation from the ground is \(17^\circ\).
• The target unknown is the height of the ramp (opposite side), which we will denote as \(h\).

Apply right triangle trigonometry

Using the Right Triangle Trigonometry concept, we relate the opposite side and the hypotenuse using the sine function:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]
Using the Angle of Elevation of \(17^\circ\):
\[
\sin(17^\circ) = \frac{h}{18}
\]

Solve for the height

Isolating the variable \(h\):
\[
h = 18 \cdot \sin(17^\circ)
\]

Calculate the numerical value

Using a calculator to find the value of \(\sin(17^\circ)\):
\[
\sin(17^\circ) \approx 0.29237
\]
Multiply by \(18\):
\[
h \approx 18 \cdot 0.29237 \approx 5.2627\text{ inches}
\]

Round to the nearest tenth

Rounding \(5.2627\) to the nearest tenth of an inch:
\[
h \approx 5.3\text{ inches}
\]
</reasoning>

<answer>
\(5.3\)
</answer>

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