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Related Rates and plane

  • Thread starter Weave
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  • #1
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Homework Statement


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?

Homework Equations


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

The Attempt at a Solution


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

  • #2
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c by the way is the hypotnuse, a in the altitude, and b is length the plane travels,
 
  • #3
HallsofIvy
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Homework Statement


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?

Homework Equations


Law of Cosines:
[tex]c^2=a^2+b^2-2abCos(\theta)[/tex]
[tex]a=11km[/tex]
[tex]b=4km[/tex]
b is NOT "4km". b is a variable and you are told that db/dt= 4 km/min

[tex]\frac{da}{dt}=0[/tex]
[tex]\frac{db}{dt}=4km/min[/tex]

The Attempt at a Solution


First using the law of cosines I found c at that particular moment.
[tex]c=\sqrt(137-88Cos(23\pi/36))[/tex]
Second I found the derivitive of the law of cosines
Working everything out I get:
[tex]\frac{dc}{dt}=\frac{16-44cos(23\pi/36)+44sin(23\pi/36)}{c}[/tex]
I plug in c and get the wrong answer, what did I do wrong?
Also, there is no reason to convert 25 degrees to [itex]23\pi/36[/itex] 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!).
 
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  • #4
143
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Homework Statement


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?

Homework Equations


Law of Cosines:
[tex]c^2=a^2+b^2-2abCos(\theta)[/tex]
[tex]a=11km[/tex]
[tex]b=4km[/tex]
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 [itex]23\pi/36[/itex] 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!).
But at that instant isn't b=4km?
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
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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 "[itex]sin(23\pi/36)[/itex]"? You don't differentiate the cosine- its a constant.

The law of cosines tell you that
[tex]c^2= 11^2+ b^2- 22b cos(115)[/tex]
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
143
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ah! thanks!
 

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