Let \( d \) be the distance Ray is from home at the time when Rosa has traveled 7 miles north. Since Ray works due east of home, we can represent their positions using a right triangle where:
- Rosa's position is at (0, 7) since she is traveling north.
- Ray's position is at (d, 0), where \( d \) is the distance Ray has traveled east.
According to the problem, the distance between Rosa and Ray is 1 mile more than Ray's distance from home. The distance between them is given by the formula for the distance between two points in a Cartesian plane:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in their coordinates:
\[ \text{Distance} = \sqrt{(d - 0)^2 + (0 - 7)^2} = \sqrt{d^2 + 49} \]
The problem states that this distance is one mile more than Ray's distance from home, which can be written as:
\[ \sqrt{d^2 + 49} = d + 1 \]
Next, we will square both sides to eliminate the square root:
\[ d^2 + 49 = (d + 1)^2 \]
Expanding the right side:
\[ d^2 + 49 = d^2 + 2d + 1 \]
Now, we can simplify by subtracting \( d^2 \) from both sides:
\[ 49 = 2d + 1 \]
Next, we isolate \( d \):
\[ 49 - 1 = 2d \] \[ 48 = 2d \] \[ d = 24 \]
Thus, Ray is 24 miles from home.
To summarize, Ray is 24 miles from home.