Question

At 3pm, Brunhilde headed north at 30 kilometres per hour. Two hours later Ludwig headed south at 40 kilometres per hour. At what time will they be 340 kilometres apart?

Answer 1

To solve this problem, we can analyze the situation step by step.

**Step 1: Calculate the distance each person travels.**

Let's denote:
- \( t \) as the time in hours since Brunhilde started traveling.

Brunhilde starts at 3 PM and travels north at a speed of 30 km/h. Therefore, the distance she has traveled after \( t \) hours is:

\[
\text{Distance}_{\text{Brunhilde}} = 30t \text{ km}
\]

Ludwig starts 2 hours later at 5 PM and travels south at a speed of 40 km/h. Thus, Ludwig travels for \( t - 2 \) hours (because he starts traveling 2 hours after Brunhilde). Therefore, the distance he has traveled after \( t \) hours is:

\[
\text{Distance}_{\text{Ludwig}} = 40(t - 2) \text{ km}
\]

**Step 2: Set up the equation.**

We want to find when they are 340 km apart. The total distance between them when they are traveling in opposite directions is the sum of the distances they traveled.

So, we set up the equation:

\[
30t + 40(t - 2) = 340
\]

**Step 3: Simplify the equation.**

Now, we can simplify the left side:

\[
30t + 40t - 80 = 340
\]

Combine like terms:

\[
70t - 80 = 340
\]

**Step 4: Solve for \( t \).**

Add 80 to both sides:

\[
70t = 420
\]

Now, divide both sides by 70:

\[
t = 6
\]

**Step 5: Find the actual time.**

Since \( t \) represents the number of hours since Brunhilde started, and she started at 3 PM, we add 6 hours to 3 PM:

\[
3 \text{ PM } + 6 \text{ hours } = 9 \text{ PM}
\]

Thus, Brunhilde and Ludwig will be 340 kilometers apart at **9 PM**.