I just need help getting started. I can't figure out the equation I need to use!
An airplane passes over an airport at noon traveling 500 mi/hr due
West. At 1:00 pm, another airplane passes over the same airport at the same
elevation traveling due North at 550 mi/hr. Assuming both planes maintain their
(equal) elevations, how fast is the distance between them changing at 2:30 pm?
draw the figure, and start with the pythoragus theorem.
Ok so I figured the equation out but after I took the derivative I wasn't sure what to plug in where
To solve this problem, we need to use the concept of rates of change and apply it to the given scenario. The distance between the airplanes is changing over time, so we need to find the rate at which this distance is changing at 2:30 pm.
Let's break down the problem and determine the information we have:
1. The first airplane is traveling due West at a speed of 500 mi/hr.
2. The second airplane is traveling due North at a speed of 550 mi/hr.
3. The time we are interested in is 2:30 pm.
To find the rate at which the distance between the two airplanes is changing, we can use the concept of relative motion. Since the two airplanes are moving in perpendicular directions (West and North), their velocities can be treated as the sides of a right triangle.
First, we need to calculate the distance each airplane has traveled by 2:30 pm:
The first airplane has traveled for 2.5 hours at 500 mi/hr, so it has traveled a distance of 2.5 hours * 500 mi/hr = 1250 miles due West.
The second airplane has traveled for 2.5 hours at 550 mi/hr, so it has traveled a distance of 2.5 hours * 550 mi/hr = 1375 miles due North.
Next, we have a right triangle formed by the two airplanes and the distance between them at 2:30 pm. The sides of this right triangle are the distances traveled by each airplane (1250 miles and 1375 miles).
At this point, we have a right triangle with known side lengths. We can use the Pythagorean theorem to find the distance between the two airplanes at 2:30 pm:
Distance^2 = (1250 miles)^2 + (1375 miles)^2
Now we have the total distance. To find the rate at which the distance between the two airplanes is changing, we need to take the derivative with respect to time. This will give us the rate of change of the distance at 2:30 pm.
Once we obtain the derivative of the equation, we can substitute the values into the equation to find the rate of change at 2:30 pm.