Airplane Angle Rate-of-Change Problem  TOPIC_SOLVED

Limits, differentiation, related rates, integration, trig integrals, etc.

Airplane Angle Rate-of-Change Problem

Postby jaybird0827 on Wed May 27, 2009 12:01 am

This airplane is flying at 500 km/h in a straight direction at a constant 5000 m. is the angle of elevation from a fixed point of observation. What is the rate of change of when it is 30°? Required to answer in DMS/s and round to one decimal place.

Solution:

Drew the right triangle as follows: Vertical segment labeled 5000 m = 5 km, assigned variable hj. At the top, a stick figure of a jet plane. Horizontal segment labeled 5 km, assigned variable b. Hypotenuse labeled 10 km, assigned variable c. The 5 and the 10 we get from the 30-60-90 relationship. Angle of elevation is angle , labeled = 30 deg (using degree symbol).

We also know that = 500 km/h and = 0. So,

tan = h/b and

sec2 d/dt = (b dh/dt - h db/dt)/h2

Plugging in,
sec2 d /dt = (0 - 5*500)/25 = -100. But sec = 10/5sqrt3, then sec2 = 4/3 and
4/3 d /dt = -100. Multiplying both sides by 3/4 gives d /dt = -75; i.e. -75 radians/h.

Using a calculator, I get -75 /h * 1 h / 60 min * 1 min / 60 s = approx -0.0208333333 / s which converts to -1.19+°, or (-1 deg 11 min 37.2 sec) /s , answer
User avatar
jaybird0827
 
Posts: 24
Joined: Tue May 26, 2009 6:31 pm
Location: NC

Sponsor

Sponsor
 

Postby stapel_eliz on Wed May 27, 2009 1:13 am

I think you swapped "h" and "b" in the denominator when you took the derivative.





:wink:
User avatar
stapel_eliz
 
Posts: 1720
Joined: Mon Dec 08, 2008 4:22 pm

Re: Airplane Angle Rate-of-Change Problem

Postby jaybird0827 on Fri May 29, 2009 7:35 pm

Eliz,

Appreciate your attention to details. Hmm, let's see, "Low d high minus high d low, over the square of what's below") -

,

And here's where I made my error - I swapped the "square of "what's below" with the square of what was above ... :oops:
Correction, should be

, not , in the denominator,



The rest of the fix should be easy.



, and







Using a calculator, I get



, or

Answer:

Does this make sense?
User avatar
jaybird0827
 
Posts: 24
Joined: Tue May 26, 2009 6:31 pm
Location: NC

  TOPIC_SOLVED

Postby stapel_eliz on Sat May 30, 2009 1:39 pm

jaybird0827 wrote:


Why did the sign change? (The angle should be getting smaller, shouldn't it?)

Check your division: 100/4 = 25, not 50. :oops:
User avatar
stapel_eliz
 
Posts: 1720
Joined: Mon Dec 08, 2008 4:22 pm

Re: Airplane Angle Rate-of-Change Problem

Postby jaybird0827 on Mon Jun 01, 2009 2:44 pm

:oops:
Wow, I must have been in "la la land". So, further corrections








, or

Answer:

I definitely agree it would have to be negative, and at that, a very small change per second.

Thanks for your help and your patience!
User avatar
jaybird0827
 
Posts: 24
Joined: Tue May 26, 2009 6:31 pm
Location: NC

Postby stapel_eliz on Mon Jun 01, 2009 2:48 pm

Looks good to me! :thumb:
User avatar
stapel_eliz
 
Posts: 1720
Joined: Mon Dec 08, 2008 4:22 pm


Return to Calculus

cron