A man is walking to a town at 6.5 km/ h. He starts at a point 20 km from the town.
a Draw a graph of distance travelled against time taken.
b State the gradient and vertical axis intercept.
c Give a formula for the distance travelled d km after t hours.
d If the man starts walking at 3 p.m. and sunset is at 5:45 p.m., will he arrive during daylight

Let's break down the problem step by step:

### Part (a) - Graph of Distance Traveled Against Time Taken

First, we will consider the relationship between distance traveled and time taken. The man is walking at a constant speed of 6.5 km/h.

- The horizontal axis will represent time \( t \) in hours.
- The vertical axis will represent the distance traveled \( d \) in kilometers.

The man starts 20 km away from the town, so at \( t = 0 \), \( d = 0 \). As time progresses, the distance he has traveled will increase linearly.

### Part (b) - Gradient and Vertical Axis Intercept

- **Gradient**: The gradient of the line is the speed of the man, which is \( 6.5 \) km/h.
- **Vertical axis intercept**: Since we start measuring distance from the point where the man starts walking, the vertical intercept is \( 0 \) km.

### Part (c) - Formula for the Distance Traveled \( d \) km After \( t \) Hours

The formula for the distance traveled after \( t \) hours, given his speed is 6.5 km/h, is:

\[
d(t) = 6.5t
\]

Where:
- \( d(t) \) is the distance traveled after \( t \) hours.

### Part (d) - Will He Arrive During Daylight?

Now, let's determine if he will arrive during daylight.

- **Starting time**: 3:00 p.m.
- **Sunset time**: 5:45 p.m.
- **Total time before sunset**: From 3:00 p.m. to 5:45 p.m. is 2 hours and 45 minutes, or \( 2.75 \) hours.

The time required to reach the town can be found by setting \( d(t) \) equal to the total distance (20 km):

\[
20 = 6.5t
\]

Solving for \( t \):

\[
t = \frac{20}{6.5} \approx 3.08 \text{ hours}
\]

3.08 hours is approximately 3 hours and 5 minutes.

**Time of arrival**: 3:00 p.m. + 3 hours 5 minutes = 6:05 p.m.

Since sunset is at 5:45 p.m., he will arrive **after** sunset, meaning he will not arrive during daylight.

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### Graphical Representation
If you want, I can create a graph based on this data.

### Summary:
- **Gradient**: 6.5 km/h
- **Vertical Intercept**: 0 km
- **Formula**: \( d(t) = 6.5t \)
- **Arrival Time**: After sunset

Would you like a graph or any more details?

### Related Questions:
1. How long would it take if he increased his speed to 8 km/h?
2. If the sunset was at 6:30 p.m., would he arrive before sunset?
3. What would be the distance covered after 1.5 hours?
4. What is the total distance he would have traveled by 7:00 p.m. if he kept walking?
5. How would the graph change if he started 25 km away?

### Tip:
Always break down problems into smaller steps, especially when dealing with time and distance relationships.

A man is walking to a town at 6.5 km/h. He starts at a point 20 km from the town.
a) Draw a graph of distance travelled against time taken.
b) State the gradient and vertical axis intercept.
c) Give a formula for the distance travelled d km after t hours.
d) If the man starts walking at 3 p.m. and sunset is at 5:45 p.m., will he arrive during daylight?

Solve a real-world distance-time graph problem involving a man walking at 6.5 km/h from 20 km away. This problem covers linear equations, calculating distance over time, and determining whether he arrives before sunset. Suitable for Grades 7-9.

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