Kate and Jim exercise at their local beach, going from one end to the other and back. Starting together, Kate jogs at a steady pace of 10km per hour, while Jim walks at 6km per hour. They first meet again after 30 minutes.

How long, in kilometres, is the beach?

Let's denote the length of the beach as \( L \) kilometers.

Kate jogs at a steady pace of 10 km per hour, and Jim walks at a pace of 6 km per hour. They first meet again after 30 minutes.

First, convert 30 minutes to hours:
\[ 30 \text{ minutes} = \frac{30}{60} \text{ hours} = 0.5 \text{ hours} \]

In 0.5 hours, the distances covered by Kate and Jim can be calculated as follows:
- Distance covered by Kate: \( 10 \text{ km/h} \times 0.5 \text{ hours} = 5 \text{ km} \)
- Distance covered by Jim: \( 6 \text{ km/h} \times 0.5 \text{ hours} = 3 \text{ km} \)

Since they start together and meet after traveling for 0.5 hours, their combined distance (5 km by Kate and 3 km by Jim) should add up to the length of the beach times 2 (since they go from one end to the other and back). Thus:
\[ 5 \text{ km} + 3 \text{ km} = 2L \]

Therefore:
\[ 8 \text{ km} = 2L \]
\[ L = \frac{8}{2} \]
\[ L = 4 \text{ km} \]

So, the length of the beach is \( 4 \) kilometers.

Would you like further details or have any questions?

Here are 8 related questions to further explore the topic:

1. If Kate increased her jogging speed to 12 km/h, how soon would they meet again?
2. How long would it take for Kate and Jim to meet if Jim started 10 minutes after Kate?
3. What is the total distance covered by Kate when they first meet again?
4. How long would it take for Kate to complete one round trip along the beach?
5. If the beach were 6 kilometers long, how long would it take for them to meet again?
6. How much farther does Kate travel compared to Jim when they first meet again?
7. How many times do Kate and Jim meet if they both exercise for 2 hours?
8. What is the ratio of the time taken for Kate to travel one way to the time taken for Jim?

**Tip:** To solve problems involving relative speeds and meeting points, always consider the combined distance traveled by both individuals or objects to find the total distance or time taken.

Explore how to calculate the length of a beach using speed and time in this problem involving Kate jogging and Jim walking. Understand their meeting point after 30 minutes, with step-by-step calculations. Suitable for Grades 7-9.

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