Final answer:
The distance between the rocket and the tracking station is changing at a rate of 1.6 miles per second when the rocket is 4 miles up. This is found using the Pythagorean theorem and related rates.
Step-by-step explanation:
To find how fast the distance between the rocket and the tracking station is changing when the rocket is 4 miles up, we can employ the Pythagorean theorem. We're given that the rocket is traveling at a speed of 2 miles per second, and we need to determine the rate of change of the distance between the rocket and the radar station at a particular moment.
Let's denote the distance from the launch pad to the radar station as x (which is 3 miles) and the height of the rocket as y at any given time.
The distance from the rocket to the radar station at any time can be represented as z. According to the Pythagorean theorem, we have z^2 = x^2 + y^2. When y is 4 miles, z can be calculated as:
z = √(x^2 + y^2)
= √(3^2 + 4^2)
= √(9 + 16)
= √25
= 5 miles
Since the rocket's vertical speed is constant at 2 miles per second, we can calculate the rate of change of the distance z with respect to time using related rates:
dz/dt = (dz/dy) * (dy/dt) = (1/(2*z)) * 2y * dy/dt
Plugging in the values, we get:
dz/dt = (1/(2*5)) * 2*4 * 2
= (1/10) * 8 * 2
= 1.6 miles per second
Therefore, the distance between the rocket and the tracking station is changing at a rate of 1.6 miles per second when the rocket is 4 miles up.