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1. Find the angle between the asymptotes to the hyperbola 3x² - 5xy -2y² + 17x + y + 14 = 0.
2. Two sides of a triangle are 4 meters and 5 meters in length and the angle between them is increasing at the rate of 0.06 radians/second. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length if
.3. If
,4. Solve:
.5. Find the equation of the ellipse whose foci are (4,0) and (-4,0) and
.6. For what values of x is the rate of increase of
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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Ganesh, can you make a question bank for younger kids?
I'll be here at least once every decade.
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Hi ganesh
For the second problem:
Assume that the angle between the sides of fixed length is θ, so
A = ½(4)(5) sin θ=10 sin θ where A is the area of triangle at any time , so
dA/dt =10 cos θ (dθ/dt) when θ=pi/3 and dθ/dt = 0.06
dA/dt =10 cos (pi/3) * 0.06
=10 (½)(0.06)=0.3
Best Regards
Riad Zaidan
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Hi ganesh
For the third problem:
w=x+2y+z^2 , x= cos(t) , y = sin(t) , z=t
dw/dt dw/dx)(dx/dt)+ (dw/dy)(dy/dt)+( dw/dz)(dz/dt)
=1(-sin (t) + 2(cos (t) + 2 z (1)
= -sin(t)+2 cos(t) + 2z
Best Regards
Riad Zaidan
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Hi ganesh
For the forth problem:
dy/dx + xy =x
dy/dx=x-xy=x(1-y)
dy/(1-y )=x dx
∫dy/(1-y )=∫x dx
-ln(1-y) x²) /2 + c
ln(1/(1-y))=( x²) /2 + c ⇒
1/(1-y)=e^(( x²) /2 + c) ⇒
1-y = 1/(e^(( x²) /2 + c))
y=1-1/(e^(( x²) /2 + c)) and you can simplify more
Best Regards
Riad Zaidan
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Hi ganesh;
We were working on this one at the same time but Riad was faster.
For #4;
Use separation of variables.
Integrate both sides:
Raise both sides to the power of e.
c= e^c
Invert both sides and solve for y:
Solution is:
Last edited by bobbym (2009-08-15 22:26:56)
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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Hi ganesh;
For the sixth problem:
Assume that
y= x^3-5x^2+5x+8 and differentiate both sides w.r.t (t) we get:
dy/dt=(3 x^2 - 10 x + 5) (dx/dt) ...........(1)
but dy/dt =2 * (dx/dt) so substitute in (1) we have the following:
2 (dx/dt) =(3 x^2 - 10 x + 5) (dx/dt) so if dx/dt ≠0 we get
2 = 3 x^2 - 10 x + 5 therfore
3 x^2 - 10 x + 3 = 0 so
(3x-1)(x-3)=0 so
either x= 1/3 or x= 3 Q.E.D
Best Regards
Riad Zaidan
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Hi ganesh;
For the fifth problem:
The foci are (4,0) and (-4,0) and e=1/3 the foci are on the x-axis with center on (0,0)
c = 4 but e=c/a so c/a= 1/3 and we have
4/a=1/3 so a =12 but c^2 = a^2 - b^2 or b^2=a^2-c^2=144-16=128
so the requiered equation is
(x^2)/144 + (y^2)/128 = 1
Best Regards
Riad Zaidan
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1. Find the angle between the asymptotes to the hyperbola 3x² - 5xy -2y² + 17x + y + 14 = 0.
This is honestly the most lengthy, involved problem I've ever worked on. I never studied the general conic equation while in school, so this is new for me. Hopefully I've done everything correctly.
First we need to get rid of the xy term by doing a rotation of axes..
So the slope of the asymptotes are
The hyperbola is oriented parallel to the y-axis and to get the angle between the asymptote and x-axis we use
So the angle between the asymptotes is
Last edited by Fruityloop (2009-09-10 21:36:40)
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