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This problem involves two boats, A and B, which start at positions \(P\) and \(Q\) respectively, with a distance of 36 km between them. After starting at the same time, boat A travels at a speed of 35 km/h and boat B at 50 km/h. They are moving in perpendicular directions.

Let's break down the steps to solve each part:

**(a) Calculate the distance traveled by boat A:**

Boat A travels for 90 minutes, which is 1.5 hours. Using the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
\[
\text{Distance by boat A} = 35 \, \text{km/h} \times 1.5 \, \text{h} = 52.5 \, \text{km}
\]

**(b) Calculate the distance traveled by boat B:**

Similarly for boat B:
\[
\text{Distance by boat B} = 50 \, \text{km/h} \times 1.5 \, \text{h} = 75 \, \text{km}
\]

**(c) Calculate the distance between the two boats after 90 minutes:**

After 90 minutes, the two boats form a right triangle where the distance they traveled are the two legs of the triangle, and the original distance between them forms one leg of the triangle. Using the Pythagorean theorem:
\[
\text{Distance}^2 = 36^2 + 52.5^2 + 75^2
\]
Let's calculate this final distance.The distance between the two boats after 90 minutes is approximately 98.37 km.

Would you like more details on this solution, or do you have any questions?

Here are five related questions to explore further:
1. How would the problem change if boat A's speed was 40 km/h instead of 35 km/h?
2. What would the total time be for the boats to be 120 km apart if they kept traveling at the same speeds?
3. How would the distance between the boats change after 2 hours instead of 90 minutes?
4. How can we use vector addition to represent the boats' motion?
5. How would we solve this problem if the boats were traveling in non-perpendicular directions?

**Tip:** When dealing with distances in motion, always ensure that time units are consistent with the speed to avoid errors.

兩艘船A和B原本分別位於位置P和Q上，它們之間的距離是36km。如圖所示，它們同時起航但以不同方向航行。船A和船B分別以固定速率35km/h和50km/h航行。過了90分鐘後，計算船A所走的距離、船B所走的距離，以及它們之間的距離。

Learn how to calculate the distance between two boats moving in perpendicular directions with speeds of 35 km/h and 50 km/h, using the Pythagorean theorem. This detailed explanation is suitable for students in Grades 10-12.

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