Applications of Calculus to the physical world (1 Viewer)

[DavidC]

Two straight roads meet at an angle of 60 degrees. A starts from the intersection and travels along one road at 40 km/h. One hour later B starts from the intersection and travels along the other road at 50 km/h. At what rate is the distance between them changing three hours after A starts?

Can someone help me out with this question?? and also explain if possible.
Thanx

 

[Fosweb]

It's a rate of change question...

First work out what each side of the triangle is in terms of time (call it 't')
So one side will be 40t (as that's how far it will have gone in 't' hours), and the other side will be 50t - 50 (cause it is always one hour behind - and it goes 50 kms in one hour.)

Now you need to find the distance they are separated by: Use cosine rule, because you have two sides and the angle between them. Distance = D

D^2 = A^2 + B^2 - 2ABcos60
D^2 = A^2 + B^2 - AB

D^2 = (40t)^2 + (50t - 50)^2 - (40t)(50t - 50)
D^2 = 1600t^2 + 2500t^2 - 5000t + 2500 - 2000t^2 + 2000t
D^2 = 2100t^2 - 3000t + 2500

So: D = SQRT(2100t^2 - 3000t + 2500)

Now you need the rate at which it is changing with time, so differentiate the distance they are separated by with respect to time: getting:

dD/dt = 1/2(2100t^2 - 3000t + 2500)^(-1/2).(4200t - 3000)
dD/dt = (2100t - 1500)[(2100t^2 - 3000t + 2500)^(-1/2)]
dD/dt = (2100t - 1500) / (SQRT(2100t^2 - 3000t + 2500))

You need this rate at t = 3 hours

@ t = 3,
dD/dt = 4800 / SQRT(12400)
dD/dt = 43.11 km h^(-2) (km/h/h)

Hopefully thats right...
Need more explanation?

 

[underthesun]

First, draw a graph of x and y axis


Y
|-----/ A(x, y)
|----/
|---/
|--/
|-/ 60
|/ B(a, 0)
O``````````````````> X

now, that 60 degrees is the angle of intersection.

The rate of change of the distance of A from the centre is 40 km/h.
The rate of change of A's x co-ordinate is 40(cos60)km/h

dx/dt = 40 (cos60)

The rate of change of A's y co-ordinate is 40(sin60)km/h

dy/dt = 40 (sin60)

The rate of change of a (B's x co-ordinate) is 50

da/dt = 50

Now, the distance between the two points are

D = sqrt( (x-a)^2 + y^2 )

D^2 = (x - a)^2 + y^2

D^2 = x^2 - 2ax + a^2 + y^2

2D (dD/dt) = 2x (dx/dt) + 2a (da/dt) + 2y (dy/dt) - 2a (dx/dt) - 2x (da/dt)

the above line used product rule : d(uv)/dx = u(dv/dx) + v(du/dx)

now that you've got the equation above, find out D (distance between A and B when it's 3 hours. A would be 120 km down one road while B would be 100 km down the other road.

hence,

x = 120 (cos60)
y = 120 (sin60)
a = 100

find D, (easy), and substitude all the value (including the dy/dxs) into the equation
2D (dD/dt) = 2x (dx/dt) + 2a (da/dt) + 2y (dy/dt) - 2a (dx/dt) - 2x (da/dt)

and tell me if the answer is right

 

[underthesun]

Boo, it took me more than 4 minutes to type that up?

Yep, i got 43.10527249, which is the same answer as fosweb's

 

[Fosweb]

but is it right?

i've seen these questions coming towards a point from a certain distance away - but this is the first time i have seen one going away from the point. What book is it from?

 

[DavidC]

Hey thanx alot!!
yeh the answer's 43.1 km/h

thanx for the explanation.
the textbook is New Senior Mathematics 3unit by Fitzpatrick.

 

[freaking_out]

hey, have u guys seen questions like this in the hsc?????i don't know...it looks strange.

 

[Fosweb]

I have seen cones and boxes and water flowing into cones and those type of rates questions, but havent really looked that much at these ones, as they seem simpler.
But I suppose they could ask them if they want - they are just rates questions.

 

[freaking_out]

Originally posted by Fosweb
I have seen cones and boxes and water flowing into cones and those type of rates questions, but havent really looked that much at these ones, as they seem simpler.
But I suppose they could ask them if they want - they are just rates questions.

yeah we haven't seen these types...so its only fitzpatrick who asks those sort of questions.

 

[Fosweb]

This years Maths in Focus has a heap of rates questions. Have a look at them if you can.

 

[freaking_out]

Originally posted by Fosweb
This years Maths in Focus has a heap of rates questions. Have a look at them if you can.

na, i have cambridge...which is more than enough...

 

[enak]

Cambridge is hectic, extension questions are hard.

 

[freaking_out]

Originally posted by enak
Cambridge is hectic, extension questions are hard.

yeah, true u get nightmares over them!!!