# Related Rates and plane

1. Apr 1, 2007

### Weave

1. The problem statement, all variables and given/known data
This last related rates HW problem is givin me trouble for some odd reason.
A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 11 km and climbs at an angle of 25 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 1 minutes later?

2. Relevant equations
Law of Cosines:
$$c^2=a^2+b^2-2abCos(\theta)$$
$$a=11km$$
$$b=4km$$
$$\frac{da}{dt}=0$$
$$\frac{db}{dt}=4km/min$$

3. The attempt at a solution
First using the law of cosines I found c at that particular moment.
$$c=\sqrt(137-88Cos(23\pi/36))$$
Second I found the derivitive of the law of cosines
Working everything out I get:
$$\frac{dc}{dt}=\frac{16-44cos(23\pi/36)+44sin(23\pi/36)}{c}$$
I plug in c and get the wrong answer, what did I do wrong?

Last edited: Apr 1, 2007
2. Apr 2, 2007

### Weave

c by the way is the hypotnuse, a in the altitude, and b is length the plane travels,

3. Apr 2, 2007

### HallsofIvy

b is NOT "4km". b is a variable and you are told that db/dt= 4 km/min

Also, there is no reason to convert 25 degrees to $23\pi/36$ since it is a constant. That doesn't change the result but I thought it was peculiar to convert from degrees to radians (and surprised that it was such a simple result!).

Last edited by a moderator: Apr 2, 2007
4. Apr 2, 2007

5. Apr 2, 2007

### HallsofIvy

Oops! Yes, I skipped over the "1 minute later" part.

However the point is that is not a "constant"- b is changing as time goes on. You cannot evaluate at b= 4 until after you take the derivative.
And how did you get that "$sin(23\pi/36)$"? You don't differentiate the cosine- its a constant.

The law of cosines tell you that
$$c^2= 11^2+ b^2- 22b cos(115)$$
Differentiating that with respect to t gives you
[tex]2c dc/dt= 2b db/dt- 22cos(115) db/dt[/itex]

Now use the fact that, at this instant, b= 4 km, db/dt= 4 km/min. You will need to determine c, at this instant, from the law of cosines.

6. Apr 2, 2007

ah! thanks!