an air traffic controller spots two planes at the same altitude flying towards one another. their flight paths form a right angle at point p. One plane is 150 miles from point p and is moving 450mph. the other plane is moving at 450mph but is 200 miles from point p. write the distance d between the planes as a function of time t.
The first plane will reach p in 150/450 hours or 20 minutes. At that point, the second plane will be 50 miles away from point p.
Does this help? Thanks for asking.
I rellu don't noe
To write the distance, "d," between the planes as a function of time, "t," we can use the Pythagorean theorem. Let's break down the problem step by step:
1. As the planes are flying towards each other, the distance between them decreases over time. Let's consider their positions at a specific time, t.
2. The first plane is 150 miles from point P and moving towards it at a speed of 450 mph. So, the position of the first plane after time t can be represented as 150 - 450t.
3. Similarly, the second plane is 200 miles from point P and moving towards it at a speed of 450 mph. Therefore, the position of the second plane after time t can be represented as 200 - 450t.
4. Now, we can use the Pythagorean theorem to find the distance, d, between the two planes at time t. According to the theorem, the square of the hypotenuse (d^2) is equal to the sum of the squares of the other two sides.
5. In this case, the distance d squared is equal to the square of the difference between the positions of the planes:
d^2 = (150 - 450t)^2 + (200 - 450t)^2
6. To simplify the equation, we can expand and combine like terms:
d^2 = 450^2t^2 - 2(150 + 200)450t + 150^2 + 200^2
d^2 = 450^2t^2 - 2(350)450t + 150^2 + 200^2
d^2 = 202,500t^2 - 315,000t + 65,000
7. Finally, the function for the distance, d, between the two planes as a function of time, t, is:
d(t) = sqrt(202,500t^2 - 315,000t + 65,000)
Note: Taking the square root gives us the actual distance, but remember to consider the positive value since distance is always positive.