A balloon rises at a rate of 4 meters per second from a point on the ground 40 meters from an observer. What is the rate of change of the angle of elevation (in rad/sec) when the balloon is 40 meters above the ground?

Question

anonymous · Answer

|dw:1352005415021:dw|

anonymous · Answer

that's the equation relating y and theta...

anonymous · Answer

What does that equation look like?

anonymous · Answer

differentiate with respect to time, solve for dtheta/dt

anonymous · Answer

$$\tan \theta =\frac{ y }{ 40}$$

anonymous · Answer

differentiate and solve for d theta / dt..
plug in values for y, y' and theta...

anonymous · Answer

think you got it now?

anonymous · Answer

Can you please show me the set up of this problem because I do not understand.

anonymous · Answer

you're doing related rates,  you've done a few problems where you differentiate with respect to time?  using the chain rule?

anonymous · Answer

should be familiar...

anonymous · Answer

Yes I have but I've never taken Trig...

anonymous · Answer

eg.  if x and y are functions of time...

then ( x^2 +y^2 ) ' = 2x*dx/dt + 2y*dy/dt       
etc

anonymous · Answer

so...  what's your question about trig.

anonymous · Answer

tangent is opp/adj

anonymous · Answer

Right. I don't understand where do I go from tan delta = y / 40

anonymous · Answer

don't now how you're in calc. without having taken trig. ever, but I guess there are alternate universes where anything can happen

anonymous · Answer

tan(theta)    theta is just the variable used for angles..

anonymous · Answer

it's like x

anonymous · Answer

you've done derivatives of trig functions?

anonymous · Answer

derivative of tan  is?

anonymous · Answer

Sec squared

anonymous · Answer

yes...  and using the chain rule too...:

sec^2 (theta) * d (theta) /dt

anonymous · Answer

that's the left side...

anonymous · Answer

right side is just dy/dt * 1/40

anonymous · Answer

$$\sec ^{2} \theta \frac{ d \theta }{ dt} = \frac{ dy }{ dt }\frac{ 1 }{40 }$$

anonymous · Answer

does that make sense?

anonymous · Answer

Okay. I understand that. I'm solving for dy/dt right?

anonymous · Answer

naw it asks for d theta /dt

anonymous · Answer

"What is the rate of change of the angle of elevation ..."

anonymous · Answer

find theta..  

divide both sides by sec^2(theta)

anonymous · Answer

plug in dy/dt and that's it...

anonymous · Answer

theta you find from   "What is the rate of change of the angle of elevation (in rad/sec) when the balloon is 40 meters above the ground?"    
so when y = 40m   theta = ..?

anonymous · Answer

|dw:1352006569177:dw|

anonymous · Answer

4?

anonymous · Answer

45 deg

anonymous · Answer

you should get in the habit of using radians however...
so happens that it doesn't matter in this case... but it usually matters if you're doing calculus with angles...  so  pi/4 is the angle in radians

anonymous · Answer

switch to radian mode on your calculator and find cos(pi/4)  then square it..

anonymous · Answer

k  got it?

anonymous · Answer

I got .5

anonymous · Answer

yep

anonymous · Answer

.5* dy/dt * 1/40   is d(theta) / dt

anonymous · Answer

Is that my answer?

anonymous · Answer

yes

anonymous · Answer

Okay. Thank You So Much! Sorry , for my lack of Trig knowledge making your job harder.

anonymous · Answer

no problem... is it clearer now?

anonymous · Answer

Yes. Very clear.

anonymous · Answer

cool.

anonymous · Answer

(: