Question

Rosa works due north of home. Her husband Ray works due east. They leave for work at the same time. By the time Rosa is 7 miles from​ home, the distance between them is one mile more than Ray​'s distance from home. How far from home is Ray​?

Answer 1

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.**