## Airplane Angle Rate-of-Change Problem

Limits, differentiation, related rates, integration, trig integrals, etc.
jaybird0827
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### Airplane Angle Rate-of-Change Problem

This airplane is flying at 500 km/h in a straight direction at a constant 5000 m. $\theta$ is the angle of elevation from a fixed point of observation. What is the rate of change of $\theta$ 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$\sqrt3$ km, assigned variable b. Hypotenuse labeled 10 km, assigned variable c. The 5$sqrt3$ and the 10 we get from the 30-60-90 relationship. Angle of elevation is angle $\theta$, labeled $\theta$= 30 deg (using degree symbol).

We also know that $\frac{db}{dt}$ = 500 km/h and $\frac{dh}{dt}$ = 0. So,

tan $\theta$ = h/b and

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

Plugging in,
sec2 $\theta$ d $\theta$/dt = (0 - 5*500)/25 = -100. But sec $\theta$ = 10/5sqrt3, then sec2 $\theta$ = 4/3 and
4/3 d $\theta$/dt = -100. Multiplying both sides by 3/4 gives d $\theta$/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

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

$\tan(\theta)\, =\, \frac{h}{b}$

$\sec^2(\theta)\,\frac{d\theta}{dt}\, =\, \frac{\frac{dh}{dt}\,b\, -\, h\,\frac{db}{dt}}{b^2}$

jaybird0827
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### Re: Airplane Angle Rate-of-Change Problem

Eliz,

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

$tan(\theta)\, =\, \frac{h}{b}$,

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

$\color{red}b^2$, not $b$, in the denominator,

$sec^2(\theta)\,\frac{d\theta}{dt}\, =\, \frac{\frac{dh}{dt}\,b\, -\, h\,\frac{db}{dt}}{b^2}\$

The rest of the fix should be easy.

$sec(\theta)\, =\, \frac{2}{\sqrt3}$

$\frac{4}{3}\,\frac{d\theta}{dt}\, =\, \frac{-5\,*\,500}{75} =\, \frac{-100}{3}$, and

$\frac{4}{3}\,\frac{d\theta}{dt}\, =\,\frac{-100}{3}$

$\frac{d\theta}{dt}\, =\,\frac{-100}{3}\,*\,\frac{3}{4}$

$\frac{d\theta}{dt}\, =\,50\,\frac{radians}{hr}$

Using a calculator, I get

$\frac{-50}{hr}\,*\,\frac{180}{\pi}\,*\frac{hr}{60\,min}\,*\,\frac{min}{60\,s}$

$=\,approx\,-0.795774715\,\frac{\,deg}{s}\$, or

Answer: $\,approx\,(-0\,deg\,\,47\,min\,\,44.8\,sec)\,/s$

Does this make sense?

stapel_eliz
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$\frac{d\theta}{dt}\, =\,\frac{-100}{3}\,*\,\frac{3}{4}$

$\frac{d\theta}{dt}\, =\,50\,\frac{radians}{hr}$
Why did the sign change? (The angle should be getting smaller, shouldn't it?)

Check your division: 100/4 = 25, not 50.

jaybird0827
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### Re: Airplane Angle Rate-of-Change Problem

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

$\frac{d\theta}{dt}\, =\,\frac{-100}{3}\,*\,\frac{3}{4}$

$\frac{d\theta}{dt}\, =\,-25\,\frac{radians}{hr}$

$\frac{-25}{hr}\,*\,\frac{180}{\pi}\,*\frac{hr}{60\,min}\,*\,\frac{min}{60\,s}$

$=\,approx\,-0.3978873577\,\frac{\,deg}{s}\$, or

Answer: $\,approx\,(-0\,deg\,\,23\,min\,\,52.4\,sec)\,/s$

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!

stapel_eliz
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Looks good to me!