To find the distance between Train A and Train B, we can use the distance formula. The distance formula is given as:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] are the coordinates of Train A and [tex]\((x_2, y_2)\)[/tex] are the coordinates of Train B.
Given:
- Train A is 14 miles east and 16 miles north of the depot, making its coordinates [tex]\((14, 16)\)[/tex].
- Train B is 12 miles west and 14 miles south of the depot, making its coordinates [tex]\((-12, -14)\)[/tex].
We'll substitute these coordinates into the distance formula.
1. First, calculate the difference in the x-coordinates [tex]\((x_2 - x_1)\)[/tex]:
[tex]\[ x_2 - x_1 = -12 - 14 = -26 \][/tex]
2. Next, calculate the difference in the y-coordinates [tex]\((y_2 - y_1)\)[/tex]:
[tex]\[ y_2 - y_1 = -14 - 16 = -30 \][/tex]
3. Now, square these differences:
[tex]\[ (-26)^2 = 676 \][/tex]
[tex]\[ (-30)^2 = 900 \][/tex]
4. Add these squared differences:
[tex]\[ 676 + 900 = 1576 \][/tex]
5. Finally, take the square root of the sum to find the distance:
[tex]\[ \sqrt{1576} \approx 39.698866482558415 \][/tex]
When rounded to the nearest tenth, the distance is approximately 39.7 miles.
Thus, the distance between the trains is:
Answer: A. 39.7 miles
