In the description

Question

A plane flying horizontally at an altitude of 2 mi and a speed of 410 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station. (Round your answer to the nearest whole number.)

Answer 1

If we draw a picture we see this:

The distance the plane travels is 410t where "t" is the time in hours. Then the distance "x" by the Pythagorean Theorem is:

x = √(2² +(410t)²)

x = √(4 + 168100t²)

x = (4 + 168100t²)^1/2

By using the chain rule, we can find dx/dt:

dx/dt = 1/2(4 + 168100t²)^-1/2(336200t)

But to calculate dx/dt, we need to know "t". We are given that x = 5. So using x = (4 + 168100t²)^1/2 we can find t:

5 = (4 + 168100t²)^1/2

25 = 4 + 168100t²

21 = 168100t²

21/16810 = t²

t = 0.011177 hrs

So to find dx/dt, just plug in t = 0.011177 into dx/dt = 1/2(4 + 168100t²)^-1/2(336200t)