Answer:

Step-by-step explanation:
We know that the plane travels at a <em>constant</em> speed of 14 km/min.
It passes over a radar station at a altitude of 11 km and climbs at an angle of 25°.
We want to find the rate at which the distance from the plane to the radar station is increasing 4 minutes later. In other words, if you will please refer to the figure, we want to find dc/dt.
First, let's find c, the distance. We can use the law of cosines:

We know that the plane travels at a constant rate of 14 km/min. So, after 4 minutes, the plane would've traveled 14(4) or 56 km So, a is 56, b is a constant 11. C is 90+20 or 115°. Substitute:

Evaluate:

Take the square root of both sides:

Now, let's return to our law of cosines. We have:

We want to find dc/dt. So, let's take the derivative of both sides with respect to t:
![\frac{d}{dt}[c^2]=\frac{d}{dt}[a^2+b^2-2ab\cos(C)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%5Bc%5E2%5D%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Ba%5E2%2Bb%5E2-2ab%5Ccos%28C%29%5D)
Since our b is constant at 11 km, we can substitute this in:
![\frac{d}{dt}[c^2]=\frac{d}{dt}[a^2-(11)^2-2a(11)\cos(C)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%5Bc%5E2%5D%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Ba%5E2-%2811%29%5E2-2a%2811%29%5Ccos%28C%29%5D)
Evaluate:
![\frac{d}{dt}[c^2]=\frac{d}{dt}[a^2-121-22a\cos(C)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%5Bc%5E2%5D%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Ba%5E2-121-22a%5Ccos%28C%29%5D)
Implicitly differentiate:

Divide both sides by 2c:

Solve for dc/dt. We already know that da/dt is 14 km/min. a is 56. We also know c. Substitute in these values:

Simplify:

Use a calculator. So:

And we're done!