Two planes leave an airport simultaneously. One travels north at 300 mph and the
other travels west at 400 mph. How fast is the distance between them changing after 10 minutes?
Using a scaled-up 3-4-5 triangle, the distance after 10 minutes is 1/6 (500) miles.
If z is the distance between them after t hours, then
z^2 = (300t)^2 + (400t)^2
z dz/dt = 2*300^2 t + 2*400^2 t
Now just plug in your numbers with t = 1/6
To find how fast the distance between the two planes is changing, we can use the concept of rate of change. Let's break down the problem:
We have two planes traveling in different directions, one to the north and the other to the west. This creates a right triangle, where the distance between the two planes represents the hypotenuse of the triangle, and the speeds of the planes represent the lengths of the two sides.
Let's denote the distance between the two planes as D(t), where t is the time in minutes. We can use the Pythagorean theorem to relate the distances traveled by each plane:
D(t)^2 = (300t)^2 + (400t)^2
Now, let's differentiate both sides of the equation with respect to time (t):
dD/dt = 2 * (300t) * (300) + 2 * (400t) * (400)
Simplifying further:
dD/dt = 60000t + 128000t
dD/dt = 188000t
Now, we need to find the rate of change (dD/dt) after 10 minutes, so we substitute t = 10 into the equation:
dD/dt = 188000 * 10
dD/dt = 1,880,000
Therefore, the distance between the two planes is changing at a rate of 1,880,000 miles per minute after 10 minutes.