Question

1. a ladder 5 m long leans against a wall. its foot is 1.8 m from the base of the wall. how high up the wall does the ladder reach? give your answer to 3 s.f.

hint: draw it out 2 marks

Explanation:

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<reasoning>

Identify the given values

We have a right triangle formed by the wall, the ground, and the leaning ladder.

• Hypotenuse (ladder length) \(c = 5\text{ m}\)
• Base (distance from wall) \(a = 1.8\text{ m}\)
• Height (reach up the wall) \(b\) is the unknown.

Apply the Pythagorean theorem

Using the Pythagorean Theorem for a right-angled triangle:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
1.8^2 + b^2 = 5^2
\]

Solve for the unknown height

Calculate the squares and isolate \(b^2\):
\[
3.24 + b^2 = 25
\]
\[
b^2 = 25 - 3.24 = 21.76
\]
\[
b = \sqrt{21.76} \approx 4.66476\text{ m}
\]

Round to significant figures

Using the Significant Figures knowledge point, we round the result to 3 significant figures:
\[
b \approx 4.66\text{ m}
\]
</reasoning>

<answer>
\(4.66\text{ m}\)
</answer>

<plot>
{
"elements": [
{
"type": "segment",
"params": [[0, 0], [0, 4.66]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 3,
"name": "Wall",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [1.8, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Ground",
"withLabel": true
}
},
{
"type": "segment",
"params": [[1.8, 0], [0, 4.66]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 4,
"name": "Ladder (5 m)",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Base of Wall",
"color": "#583C87",
"withLabel": false
}
},
{
"type": "point",
"params": [[1.8, 0]],
"properties": {
"name": "Foot of Ladder (1.8 m)",
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 4.66]],
"properties": {
"name": "Top of Ladder",
"color": "#F2557F",
"withLabel": true
}
}
]
}
</plot>

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Answer:

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</pre_analysis>

<reasoning>

Identify the given values

We have a right triangle formed by the wall, the ground, and the leaning ladder.

• Hypotenuse (ladder length) \(c = 5\text{ m}\)
• Base (distance from wall) \(a = 1.8\text{ m}\)
• Height (reach up the wall) \(b\) is the unknown.

Apply the Pythagorean theorem

Using the Pythagorean Theorem for a right-angled triangle:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
1.8^2 + b^2 = 5^2
\]

Solve for the unknown height

Calculate the squares and isolate \(b^2\):
\[
3.24 + b^2 = 25
\]
\[
b^2 = 25 - 3.24 = 21.76
\]
\[
b = \sqrt{21.76} \approx 4.66476\text{ m}
\]

Round to significant figures

Using the Significant Figures knowledge point, we round the result to 3 significant figures:
\[
b \approx 4.66\text{ m}
\]
</reasoning>

<answer>
\(4.66\text{ m}\)
</answer>

<plot>
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{
"type": "segment",
"params": [[0, 0], [0, 4.66]],
"properties": {
"strokeColor": "#583C87",
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"name": "Wall",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [1.8, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3,
"name": "Ground",
"withLabel": true
}
},
{
"type": "segment",
"params": [[1.8, 0], [0, 4.66]],
"properties": {
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"name": "Ladder (5 m)",
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{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Base of Wall",
"color": "#583C87",
"withLabel": false
}
},
{
"type": "point",
"params": [[1.8, 0]],
"properties": {
"name": "Foot of Ladder (1.8 m)",
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 4.66]],
"properties": {
"name": "Top of Ladder",
"color": "#F2557F",
"withLabel": true
}
}
]
}
</plot>

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