To solve the problem of finding the distance traveled by the second plane and the angle between their paths, we can follow these steps:
### Step 1: Identify Given Information
- Distance traveled by the first plane: [tex]\(282\)[/tex] km.
- Total distance apart after an hour: [tex]\(300\)[/tex] km.
### Step 2: Use the Pythagorean Theorem
The problem can be visualized as a triangle where:
- One side ([tex]\(a\)[/tex]) is the distance traveled by the first plane ([tex]\(282\)[/tex] km).
- The hypotenuse ([tex]\(c\)[/tex]) is the distance between the planes after an hour ([tex]\(300\)[/tex] km).
We want to find the distance traveled by the second plane ([tex]\(b\)[/tex]).
Using the Pythagorean Theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Substitute the known values:
[tex]\[ 282^2 + b^2 = 300^2 \][/tex]
[tex]\[ 79524 + b^2 = 90000 \][/tex]
Solve for [tex]\(b^2\)[/tex]:
[tex]\[ b^2 = 90000 - 79524 \][/tex]
[tex]\[ b^2 = 10476 \][/tex]
Taking the square root of both sides to find [tex]\(b\)[/tex]:
[tex]\[ b = \sqrt{10476} \][/tex]
[tex]\[ b \approx 102.3523 \][/tex]
So, the distance traveled by the second plane is approximately [tex]\(102.3523\)[/tex] km.
### Step 3: Use the Law of Cosines to Find the Angle
To find the angle ([tex]\(\theta\)[/tex]) between the paths of the planes, we use the Law of Cosines:
[tex]\[ \cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Substitute the known values:
[tex]\[ \cos(\theta) = \frac{282^2 + 102.3523^2 - 300^2}{2 \cdot 282 \cdot 102.3523} \][/tex]
Calculate the numerator and denominator separately:
[tex]\[ \cos(\theta) = \frac{79524 + 10476 - 90000}{2 \cdot 282 \cdot 102.3523} \][/tex]
[tex]\[ \cos(\theta) = \frac{0}{2 \cdot 282 \cdot 102.3523} \][/tex]
[tex]\[ \cos(\theta) = 0 \][/tex]
### Step 4: Determine the Angle
Since [tex]\(\cos(\theta) = 0\)[/tex], we know that:
[tex]\[ \theta = \cos^{-1}(0) \][/tex]
The corresponding angle is:
[tex]\[ \theta = 90^\circ \][/tex]
### Final Answer
The second plane has traveled approximately [tex]\(102.4\)[/tex] km, and the angle between their paths is [tex]\(90.0^\circ\)[/tex] rounded to the nearest tenth of a degree.
