Question

katherine leans a 30-foot ladder against a wall so that it forms an angle of \\(72^\circ\\) with the ground. how high up the wall does the ladder reach? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": true,
"concepts_used": [
"Right Triangle Trigonometry",
"Trigonometric Applications"
],
"new_concepts": [],
"current_concepts": [
"Right Triangle Trigonometry",
"Trigonometric Applications"
]
}
</pre_analysis>

<reasoning>

Identify given values and target

We are given:

• Length of the ladder (hypotenuse, \(h\)) = \(30\text{ ft}\)
• Angle with the ground (\(\theta\)) = \(72^\circ\)

We need to find:

• Height up the wall (opposite side, \(y\))

Set up the trigonometric ratio

Using Right Triangle Trigonometry, we relate the opposite side and the hypotenuse:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]
Using Trigonometric Applications, we substitute our known values:
\[
\sin(72^\circ) = \frac{y}{30}
\]

Solve for the height

Isolate \(y\) by multiplying both sides by \(30\):
\[
y = 30 \cdot \sin(72^\circ)
\]

Calculate the numerical value

Using a calculator to find \(\sin(72^\circ)\):
\[
\sin(72^\circ) \approx 0.9510565
\]
Multiply by \(30\):
\[
y \approx 30 \cdot 0.9510565 \approx 28.531695\text{ ft}
\]

Round to the nearest hundredth

Rounding \(28.531695\) to two decimal places:
\[
y \approx 28.53\text{ ft}
\]
</reasoning>

<answer>
\(28.53\text{ ft}\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Ground-Wall Corner",
"size": 3,
"color": "#583C87",
"withLabel": false
}
},
{
"type": "point",
"params": [[2.9, 0]],
"properties": {
"name": "Ladder Base",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 9.0]],
"properties": {
"name": "Ladder Top",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [2.9, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Ground"
}
},
{
"type": "segment",
"params": [[0, 0], [0, 9.0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Wall"
}
},
{
"type": "segment",
"params": [[2.9, 0], [0, 9.0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 4,
"name": "Ladder (30 ft)"
}
},
{
"type": "text",
"params": [[1.8, 4.8], "30 ft"],
"properties": {
"color": "#F2557F"
}
},
{
"type": "text",
"params": [[-1.2, 4.5], "Height (y)"],
"properties": {
"color": "#5583F2"
}
},
{
"type": "text",
"params": [[2.0, 0.5], "72°"],
"properties": {
"color": "#583C87"
}
}
],
"timestamps": [0.1]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Word Problem",
"knowledge_point": [
"Mathematics",
"Geometry",
"Right Triangle Trigonometry"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": true,
"concepts_used": [
"Right Triangle Trigonometry",
"Trigonometric Applications"
],
"new_concepts": [],
"current_concepts": [
"Right Triangle Trigonometry",
"Trigonometric Applications"
]
}
</pre_analysis>

<reasoning>

Identify given values and target

We are given:

• Length of the ladder (hypotenuse, \(h\)) = \(30\text{ ft}\)
• Angle with the ground (\(\theta\)) = \(72^\circ\)

We need to find:

• Height up the wall (opposite side, \(y\))

Set up the trigonometric ratio

Using Right Triangle Trigonometry, we relate the opposite side and the hypotenuse:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]
Using Trigonometric Applications, we substitute our known values:
\[
\sin(72^\circ) = \frac{y}{30}
\]

Solve for the height

Isolate \(y\) by multiplying both sides by \(30\):
\[
y = 30 \cdot \sin(72^\circ)
\]

Calculate the numerical value

Using a calculator to find \(\sin(72^\circ)\):
\[
\sin(72^\circ) \approx 0.9510565
\]
Multiply by \(30\):
\[
y \approx 30 \cdot 0.9510565 \approx 28.531695\text{ ft}
\]

Round to the nearest hundredth

Rounding \(28.531695\) to two decimal places:
\[
y \approx 28.53\text{ ft}
\]
</reasoning>

<answer>
\(28.53\text{ ft}\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Ground-Wall Corner",
"size": 3,
"color": "#583C87",
"withLabel": false
}
},
{
"type": "point",
"params": [[2.9, 0]],
"properties": {
"name": "Ladder Base",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 9.0]],
"properties": {
"name": "Ladder Top",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [2.9, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Ground"
}
},
{
"type": "segment",
"params": [[0, 0], [0, 9.0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Wall"
}
},
{
"type": "segment",
"params": [[2.9, 0], [0, 9.0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 4,
"name": "Ladder (30 ft)"
}
},
{
"type": "text",
"params": [[1.8, 4.8], "30 ft"],
"properties": {
"color": "#F2557F"
}
},
{
"type": "text",
"params": [[-1.2, 4.5], "Height (y)"],
"properties": {
"color": "#5583F2"
}
},
{
"type": "text",
"params": [[2.0, 0.5], "72°"],
"properties": {
"color": "#583C87"
}
}
],
"timestamps": [0.1]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Word Problem",
"knowledge_point": [
"Mathematics",
"Geometry",
"Right Triangle Trigonometry"
]
}
</post_analysis>