Question

toms journey is shown on the distance-time graph below.
he travelled the final 3 hours of his journey at a constant speed of \\(80\text{ km/h}\\).
work out the value of \\(d\\).

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Distance-Time Graphs",
"Constant Speed Calculation"
],
"current_concepts": [
"Distance-Time Graphs",
"Constant Speed Calculation"
]
}
</pre_analysis>

<reasoning>

Identify the given values from the graph

From the distance-time graph, we can identify the key coordinates and intervals:

• At \(t = 0\), the distance is \(0\text{ km}\).
• At \(t = 2\text{ hours}\), the distance is \(180\text{ km}\).
• From \(t = 2\text{ hours}\) to \(t = 3\text{ hours}\), the distance remains constant at \(180\text{ km}\) (Tom is stationary).
• The final part of the journey starts at \(t = 3\text{ hours}\) and ends at \(t = 6\text{ hours}\).
• The total duration of this final section is:

\[
\Delta t = 6 - 3 = 3\text{ hours}
\]

• The distance at the start of this final section (\(t = 3\)) is \(180\text{ km}\).
• The distance at the end of this final section (\(t = 6\)) is \(d\text{ km}\).

Apply the constant speed formula

We are given that Tom travelled the final 3 hours of his journey at a constant speed of \(80\text{ km/h}\).
The formula relating speed, distance, and time is:
\[
\text{Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}}
\]
Substituting the known values into the formula:
\[
80 = \frac{d - 180}{3}
\]

Solve for the unknown value d

Multiply both sides of the equation by 3 to isolate the distance term:
\[
80 \times 3 = d - 180
\]
\[
240 = d - 180
\]
Add 180 to both sides to find \(d\):
\[
d = 240 + 180
\]
\[
d = 420
\]
</reasoning>

<answer>
Work out the value of d.
\(d =\) <blank>420</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Distance-Time Graphs"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Distance-Time Graphs",
"Constant Speed Calculation"
],
"current_concepts": [
"Distance-Time Graphs",
"Constant Speed Calculation"
]
}
</pre_analysis>

<reasoning>

Identify the given values from the graph

From the distance-time graph, we can identify the key coordinates and intervals:

• At \(t = 0\), the distance is \(0\text{ km}\).
• At \(t = 2\text{ hours}\), the distance is \(180\text{ km}\).
• From \(t = 2\text{ hours}\) to \(t = 3\text{ hours}\), the distance remains constant at \(180\text{ km}\) (Tom is stationary).
• The final part of the journey starts at \(t = 3\text{ hours}\) and ends at \(t = 6\text{ hours}\).
• The total duration of this final section is:

\[
\Delta t = 6 - 3 = 3\text{ hours}
\]

• The distance at the start of this final section (\(t = 3\)) is \(180\text{ km}\).
• The distance at the end of this final section (\(t = 6\)) is \(d\text{ km}\).

Apply the constant speed formula

We are given that Tom travelled the final 3 hours of his journey at a constant speed of \(80\text{ km/h}\).
The formula relating speed, distance, and time is:
\[
\text{Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}}
\]
Substituting the known values into the formula:
\[
80 = \frac{d - 180}{3}
\]

Solve for the unknown value d

Multiply both sides of the equation by 3 to isolate the distance term:
\[
80 \times 3 = d - 180
\]
\[
240 = d - 180
\]
Add 180 to both sides to find \(d\):
\[
d = 240 + 180
\]
\[
d = 420
\]
</reasoning>

<answer>
Work out the value of d.
\(d =\) <blank>420</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Distance-Time Graphs"
]
}
</post_analysis>