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Problem statement:

A plane flying with constant speed of 4 km/min passes over a ground radar station at an altitude of 6 km and climbs at an angle of 35 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing 6 minutes later?

Hint: The law of cosines for a triangle is $c^2=a^2+b^2− 2ab \cos(\theta)$ where $\theta$ is the angle between the sides of length $a$ and $b$.


Okay, so following the typical related rates algorithm:

Identify the problem at hand by drawing a picture. I did that. It appears to me that after the plane passes over the ground station, we get an angle between the measures a and b of 90 + 35 degrees.

Awesome.

Now, let's build a table of values: $$ a = 6\,\mathrm{km}\\ a' = 0 \quad \text{(duh)}\\ b = b'\cdot t = 4\cdot 6 = 24\\ b' = 4\,\mathrm{km}/\mathrm{min}\\ t' = 6min\\ c = \sqrt{a^2 + b^2} = \sqrt{36+24} = 2\sqrt{15} \approx 2.78316\\ c' = ? $$

Okay, let's differentiate the law of cosines, which is given to us in form of a hint:

$2cc' = 2aa' + 2bb' - \frac{du}{dt}$

$ u = 2ab\cos(\theta)$

$\frac{du}{dt} = \frac{ds}{dt} \cdot \cos(\theta) - \sin(\theta)\cdot(2ab)$

$\frac{ds}{dt} = 0\cdot ab + (a'b + b'a)$

And so now we rewind: $\frac{du}{dt} = \left[ \left(0 \cdot ab + \left(a'b + b'a \right) \right) \right] \cdot \cos(\theta) - \sin(\theta) \cdot 2ab $

And so:

$2cc' = 2aa' + 2bb' - \left[ \left(0 \cdot ab + \left(a'b + b'a \right) \right) \right] \cdot \cos(\theta) - \sin(\theta) \cdot 2ab $


Boom, boom. Let's substutute what we know

$2\left(2 \sqrt{15} \right)(c') = 2(6)(0) + 2(24)(4) - \left[ \left(0\cdot (6)(24) + \left((0)(24) + (4)(6) \right) \right) \right] \cdot \cos(125^\circ) - \sin(125^\circ) \cdot 2(4)(24)$


Now what? Well, evaluate. Not sure if that's going to yield the right answer. But I don't think that it will.

Please help! I don't know how to solve this kind of problem! It seems my solution/process is incorrect!

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    $\begingroup$ Oh André, where oh where have you gone. I need to send you a fruit basket or something. I'd hate to know what kind of problems stump you for an hour. $\endgroup$ – alvonellos Oct 10 '13 at 5:10
  • $\begingroup$ What is the question? $\endgroup$ – dfeuer Oct 10 '13 at 5:13
  • $\begingroup$ I need someone to help me find out why my answer is wrong :( What's the correct way to approach this problem? $\endgroup$ – alvonellos Oct 10 '13 at 5:15
  • $\begingroup$ Where did you get 140 degrees? $\endgroup$ – Daniel McLaury Oct 10 '13 at 5:16
  • $\begingroup$ The sum of 90 + 35. So the plane is flying horizontally, then as soon as it passes over the ground station, it begins climbing at 35 degrees. So it is at a 90 degree angle, then it adds another 35 to... oh, 125 degrees! ^_^. Okay I fixed it. $\endgroup$ – alvonellos Oct 10 '13 at 5:19
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We use your notation. Let $c=c(t)$ be the distance from the plane to the radar station at time $t$. Let $b$ be the distance from the plane to the point where the climb began. Then $$c^2=36+b^2-12b\cos \theta=36+b^2+12b\sin\phi,\tag{1}$$ where $\theta=125^\circ$ and $\phi=35^\circ$. Differentiating, we get $$2c\frac{dc}{dt}=2b\frac{db}{dt}+12\sin\phi\frac{db}{dt}.\tag{2}$$ "Freeze" at time $6$ minutes after the climb began. We know everything on the right-hand side of (2), since at that instant $b=(6)(4)$.

We need to calculate $c$ at time $t=6$. Equation (1) does that, with $b=24$.

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Your solution looks okay for the most part; you've just made some mistakes. A couple points:

  1. $t'$ means the rate at which $t$ changes with respect to itself. This is of course just 1.
  2. Keep track of what's constant before breaking out the product rule, which can cause your formulae to explode. In the expression $2 a b \cos(\theta)$, both $a$ and $\cos(\theta)$ are simply constants, so it's quite easy to differentiate.
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  • $\begingroup$ Okay, so I shouldn't completely differentiate the formula before plugging in numbers? Would you mind editing my post and fixing the problem? $\endgroup$ – alvonellos Oct 10 '13 at 5:22
  • $\begingroup$ Just notice the difference between constant and non-constant. $\endgroup$ – Daniel McLaury Oct 10 '13 at 5:22
  • $\begingroup$ Okay. Fixed some numbers. $\endgroup$ – alvonellos Oct 10 '13 at 5:25
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Do you really need to use calculus?

At the moment in question, the plane is downrange from the radar station $24\cos(35^\circ)$ km at an altitude of $(6+24\cdot\sin(35^\circ)) $ km. So, $\theta$, the angle of elevation of the plane as seen from the radar station is given by:$$\theta=\tan^{-1}\frac{6+24\cdot\sin(35^\circ)}{24\cos(35^\circ)}$$ $\theta$ is obviously greater than $35^\circ$.

The plane is still climbing at $35^\circ$ at 4 km/ min. The component of its velocity along the line of sight from the radar station is:$$V_{los}=4\cdot\cos(\theta-35^\circ)$$

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  • $\begingroup$ I like this solution. $\endgroup$ – copper.hat Oct 10 '13 at 5:50
  • $\begingroup$ Of course this is really just hiding the calculus. It's the linearity of the derivative that allows us to talk about a velocity being a vector that has components in directions. $\endgroup$ – Daniel McLaury Oct 10 '13 at 5:53
  • $\begingroup$ From a Physics point of view, one might say that the calculus approach hides the physical fact about the component of velocity. $\endgroup$ – André Nicolas Oct 10 '13 at 5:57
  • $\begingroup$ @DanielMcLaury: You are correct, of course, but I think the above approach is much better at giving an intuition for the problem that is hidden by the formulaic approach in the question. Frankly, I am bewildered by the OPs approach. $\endgroup$ – copper.hat Oct 10 '13 at 5:58
  • $\begingroup$ Hey. to the person with a hammer. all the world looks like a nail... $\endgroup$ – DJohnM Oct 10 '13 at 5:59
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At the end of your table of values, $c$ isn’t $\sqrt{a^2+b^2}$; the correct formula is

$$c=\sqrt{a^2+b^2-2ab\cos\theta}\;,$$

which in this case is

$$c=\sqrt{6^2+(4t)^2-2\cdot6\cdot4t\cdot\cos 125°}=\sqrt{36+16t^2-48t\cos125°}\;.$$

The differentiation is easier if we start with the square of that, and we get $$2cc'=32t-48\cos125°\;,$$ or $$c'=\frac{16t-24\cos 125°}c\;.$$ At $t=6$ we have $c\approx 27.87813$ km and hence $c'\approx 3.93734$ km/min.

Note that with this approach you’d have been less likely to get lost in the differentiation (e.g., forgetting that $a$ and $\cos\theta$ are constants).

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  • $\begingroup$ Where did you get 135? $\endgroup$ – alvonellos Oct 10 '13 at 5:27
  • $\begingroup$ @alvonellos: Sorry: that’s a typo and should be $125°$ everywhere. Fixed now. $\endgroup$ – Brian M. Scott Oct 10 '13 at 5:28
  • $\begingroup$ Shouldn't the $\cos$ be a $\sin$? $\endgroup$ – copper.hat Oct 10 '13 at 5:29
  • $\begingroup$ @copper.hat: No; $\cos\theta$ is a constant in this calculation. $\endgroup$ – Brian M. Scott Oct 10 '13 at 5:30
  • $\begingroup$ It still says that's incorrect in the sample problem. $\endgroup$ – alvonellos Oct 10 '13 at 5:31
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This seems way too complicated. I must be missing something.

enter image description here

We have $x(t) = 4t \cos \theta$, $y(t) = 6+4t \sin \theta$. Hence the distance from the radar station is $d(t)^2 = x(t)^2 + y(t)^2 = 48\,t\,\sin\left( \theta\right) +16\,{t}^{2}+36$.

Taking the square root and differentiating gives $d'(t) = \frac{48\,{\sin}\left( \theta\right) +32\,t}{2\,\sqrt{48\,t\,\mathrm{sin}\left( \theta\right) +16\,{t}^{2}+36}}$.

Substituting $t=6$ gives: $d'(6) = \frac{48\,\sin\left( \theta\right) +192}{2\,\sqrt{288\,\sin\left( \theta\right) +612}}$.

This gives me, roughly, $d'(6) \approx 3.94$.

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  • $\begingroup$ It's not that complicated? $\endgroup$ – Michael Fulton Nov 28 '16 at 20:21
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    $\begingroup$ @MichaelFulton: I meant the OP's approach was complicated (to me). $\endgroup$ – copper.hat Nov 28 '16 at 20:25

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