Sure, let's break down the solution step-by-step.
Step 1: Define the scenario and initial speeds of the cars.
- One car travels west at a speed of 35 mph.
- The other car travels south at a speed of 20 mph.
Step 2: Calculate the distances traveled by each car after 2 hours.
- Distance traveled by the car moving west: [tex]\( \text{distance}_\text{west} = \text{speed}_\text{west} \times \text{time} = 35 \text{ mph} \times 2 \text{ hours} = 70 \text{ miles} \)[/tex]
- Distance traveled by the car moving south: [tex]\( \text{distance}_\text{south} = \text{speed}_\text{south} \times \text{time} = 20 \text{ mph} \times 2 \text{ hours} = 40 \text{ miles} \)[/tex]
Step 3: Determine the distance between the two cars after 2 hours.
- Since the cars are moving perpendicularly to each other, we can use the Pythagorean theorem.
- Let [tex]\( d \)[/tex] be the distance between the two cars.
- [tex]\( d = \sqrt{ \text{distance}_\text{west}^2 + \text{distance}_\text{south}^2 } = \sqrt{ 70^2 + 40^2 } \approx 80.6 \text{ miles} \)[/tex]
Step 4: Find the rate at which the distance between the cars is increasing.
- To find the rate at which the distance between the two cars is increasing, we use differentiation.
- Given that speed_west and speed_south are constants, the rate of increase of the distance between the two cars is given by:
- [tex]\(\text{rate of increase} = \sqrt{ \text{speed}_\text{west}^2 + \text{speed}_\text{south}^2 } = \sqrt{ 35^2 + 20^2 } \approx 40.3 \text{ mph} \)[/tex]
So, 2 hours later, the distance between the cars is increasing at a rate of approximately 40.3 mph.
