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The question you uploaded describes a scenario where two cars are approaching an intersection from different directions, and it asks to express the distance between them as a function of time.

### Given:
1. One car is 4 miles south of the intersection, moving north at 20 miles per hour.
2. The other car is 1 mile east of the intersection, moving west at 30 miles per hour.
3. At \( t = 0 \), the first car is 4 miles south, and the second car is 1 mile east.

### Let’s solve part (a):
We want to express the distance \( d(t) \) between the two cars as a function of time \( t \).

1. **Position of Car 1 (North-South)**:
   - At \( t = 0 \), Car 1 is 4 miles south.
   - Since it moves north at 20 miles per hour, its position at any time \( t \) (in hours) is given by:
     \[
     y(t) = 4 - 20t
     \]
     (where \( y(t) \) represents the distance south of the intersection).

2. **Position of Car 2 (East-West)**:
   - At \( t = 0 \), Car 2 is 1 mile east.
   - Since it moves west at 30 miles per hour, its position at any time \( t \) is given by:
     \[
     x(t) = 1 - 30t
     \]
     (where \( x(t) \) represents the distance east of the intersection).

3. **Distance between the cars**:
   The distance \( d(t) \) between the two cars can be found using the Pythagorean theorem, since the cars are traveling along perpendicular paths:
   \[
   d(t) = \sqrt{x(t)^2 + y(t)^2}
   \]
   Substituting the expressions for \( x(t) \) and \( y(t) \):
   \[
   d(t) = \sqrt{(1 - 30t)^2 + (4 - 20t)^2}
   \]

So, the distance between the two cars as a function of time is:
\[
d(t) = \sqrt{(1 - 30t)^2 + (4 - 20t)^2}
\]

Would you like a breakdown of the next steps or a detailed solution for another part of the problem?

### Relative questions:
1. How would the problem change if the speeds of the cars were different?
2. What happens to the distance as time increases? Does it increase or decrease, and why?
3. Can we find the time when the two cars are closest to each other?
4. How would we calculate the rate of change of the distance between the cars?
5. What if the cars were starting from different initial distances from the intersection?

### Tip:
Always double-check units when solving physics-based problems. Speed should correspond to the same time units as distance, such as miles per hour for distance in miles.

Two cars are approaching an intersection. One is 4 miles south of the intersection and is moving at a constant speed of 20 miles per hour. The other car is 1 mile east of the intersection and is moving at a constant speed of 30 miles per hour. Express the distance between the cars as a function of time t.

Learn how to calculate the distance between two cars approaching an intersection, one from the south and the other from the east. This math problem uses algebra, functions, and the Pythagorean theorem to express the distance as a function of time. Suitable for students in Grades 9-12.

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