Two cars leave an intersection. One car travels North, other travels East. When the car travelling North had gone 24 km, the distance between the cars was 4 km more than three times the distance travelled by the car heading East. Find the distance between two cars at theat time.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variables
Let \( x \) be the distance travelled by the car heading East.

### Step 2: Understand the distances
The car travelling North has travelled 24 km. The distance between the two cars is given as:
\[ \text{Distance} = 3x + 4 \]

### Step 3: Use the Pythagorean theorem
Since the cars are travelling at right angles to each other, we can use the Pythagorean theorem to relate the distances:
\[ (24)^2 + (x)^2 = (3x + 4)^2 \]

### Step 4: Expand the equation
Expanding the equation gives:
\[ 576 + x^2 = (3x + 4)(3x + 4) \]
\[ 576 + x^2 = 9x^2 + 24x + 16 \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ 0 = 9x^2 + 24x + 16 - x^2 - 576 \]
\[ 0 = 8x^2 + 24x - 560 \]

### Step 6: Simplify the equation
Dividing the entire equation by 8 to simplify:
\[ 0 = x^2 + 3x - 70 \]

### Step 7: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ 0 = (x + 10)(x - 7) \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
1. \( x + 10 = 0 \) → \( x = -10 \) (not valid since distance cannot be negative)
2. \( x - 7 = 0 \) → \( x = 7 \)

### Step 9: Find the distance between the two cars
Now that we have \( x = 7 \), we can find the distance between the two cars:
\[ \text{Distance} = 3x + 4 = 3(7) + 4 = 21 + 4 = 25 \text{ km} \]

### Final Answer
The distance between the two cars at that time is **25 km**.

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