Growth of an Angle
5 and 10 inches are 2 sides of a triangle. The angle between them is increasing at the rate of
5 degrees per minute. How fast is the third side of the triangle growing when the angle is 60 degrees?
(I have an answer, but no detailed solution)
5 degrees per minute. How fast is the third side of the triangle growing when the angle is 60 degrees?
(I have an answer, but no detailed solution)
posted by Zaux at
2:18 PM
20 Comments:
This post has been removed by the author.
Hi Chris ...
the published solution is expressed in inches/min.
Cosine rule:
a² = b² + c² - 2bc cosA
A angle opposite to side a.
b = 5, c = 10
Need to find da/dA. I'll treat A as being in radians to start.
2a da/dA = 2bc sinA
=> da/dA = (bc/a) sinA.
Wha A = 60º, cosA = 0.5, sinA = (√3)/2
=> a² = 25 + 100 - 2*5*10 *0.5 = 75
=> a = √(75) = 5√3
da/dA = (5*10/(5√3))*(√3)/2 = 5 inches/radian
da/dt = (da/dA)(dA/dt)
Altogether
da/dt = 5*5*π/180 inches/sec
= 0.436332... inches/sec
Deleted as had too many silly numerical errors.
But in inches/minute: 26.18 (2SF)
Chris ... they expressed it in terms of pi inches/min. ... would you like the published answer?
I goofed, I misread the question as being in inches/sec. So it's (0.436332... inches/min)
= 25π/180 inches/min
=0.138888π inches/min
(assuming no silly slips).
Chris ... don't know if you or the source is at fault, but the answers don't agree
Hi Zaux. I'll try it on a piece of paper. I did it straight into the comment editor, so may have goofed the values, but not in the method.
Okay .....
I'm sticking with 25π/180 = 5π/36 in/min.
Please post your value if that's all you've got. Thanks.
Chris ... I'm sorry ... my fault ... I was looking at your solution incorrectly.
You have the correct answer!
Phew! I thought I'd lost it for a moment. Thanks for updating so quickly.
This post has been removed by the author.
This post has been removed by the author.
... 800 should have been 8000.
Using complex numbers a, b and c to represent the sides of the triangle and let A be the angle between b and c
b = 5, c = 10 (cos[A] + i son[A])
Let a = c - b = 10 (cos[A] + i sin[A] - 5
│a│² = (10 cos[A] - 5)² + (10 sin[A])²
2│a│ d│a│/dA = 2(10 cos[A] -5)(-10 sin[A]) + 200 sin[A] cos[A]
At A = 60º = π/3, │a│² = ( 10/2 - 5)² + 75 = 75
=> │a│ = 5√3
2*5√3*d│a│/dA = 50√3 => d│a│/dA = 5
d│a│/dt = d│a│/dA * dA/dt = 5 * 5 * π/180 = 5π/36
Sorry about the deletes, too many erros for an addenda.
The third side is not growing at all, as the question gives a specific degree value - 60 degrees. This gives a specific length for the third side....
That's just me being pedantic...
I believe it does say that it is growing at a rate of 5 degrees per minute ... and asks the question, at the 60 degree point during the growth, what is the dimension of the third side?
Hi Zaux. It is growing, you're intoducing a Zeno's paradox, wherin nothing is able to move.
The length of the third side is 5√3. I had given that in both solution methods.
... on the other hand, as it is growing, it can't be at 60 degrees ;)
Just noticed it was karl who introduced the Zeno stuff.
In the Lagrangian formulation of [generalised] classical mechanics, the whole Zeno nonsense is eliminated. Length and speed are placed on an equal footing; they are treated as independent variables.
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