Question

from a 40-ft observation tower on the coast, coast guard officer nevinski sights a boat in difficulty. the angles of depression to the boat is \\(4^{\circ}\\). how far is the boat from the shoreline? round to the nearest tenth of a foot.

Explanation:

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<reasoning>

Set up the trigonometric equation

\[

$$\begin{aligned} &\theta = 4^\circ\\ &\text{Opposite} = 40\text{ ft}\\ &\tan(4^\circ) = \frac{40}{d} \end{aligned}$$

\]

Solve for the distance

\[

$$\begin{aligned} &d = \frac{40}{\tan(4^\circ)}\\ &d \approx \frac{40}{0.0699268}\\ &d \approx 572.0058\text{ ft} \end{aligned}$$

\]

Round to the nearest tenth

\[
d \approx 572.0\text{ feet}
\]
</reasoning>

<answer>
From a 40-ft observation tower on the coast, Coast Guard Officer Nevinski sights a boat in difficulty. The angles of depression to the boat is \(4^\circ\). How far is the boat from the shoreline? Round to the nearest tenth of a foot.

<blank>572.0</blank> feet
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Shoreline",
"size": 3,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 4]],
"properties": {
"name": "Tower Top",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[8, 0]],
"properties": {
"name": "Boat",
"size": 3,
"color": "#F2557F",
"withLabel": true
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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
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"Mathematics",
"Geometry",
"Trigonometric Ratios"
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Answer:

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<reasoning>

Set up the trigonometric equation

\[

$$\begin{aligned} &\theta = 4^\circ\\ &\text{Opposite} = 40\text{ ft}\\ &\tan(4^\circ) = \frac{40}{d} \end{aligned}$$

\]

Solve for the distance

\[

$$\begin{aligned} &d = \frac{40}{\tan(4^\circ)}\\ &d \approx \frac{40}{0.0699268}\\ &d \approx 572.0058\text{ ft} \end{aligned}$$

\]

Round to the nearest tenth

\[
d \approx 572.0\text{ feet}
\]
</reasoning>

<answer>
From a 40-ft observation tower on the coast, Coast Guard Officer Nevinski sights a boat in difficulty. The angles of depression to the boat is \(4^\circ\). How far is the boat from the shoreline? Round to the nearest tenth of a foot.

<blank>572.0</blank> feet
</answer>

<plot>
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{
"type": "point",
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"properties": {
"name": "Boat",
"size": 3,
"color": "#F2557F",
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