The Space Colony

[Mooch22]

The Space Colony Physica is orbiting approximately 100 km from the earth. A malfunction occrs in the computers directional system at approximately time = (Pi/2) and the colony leaves it's current orbit and begins to move along the curve y=sin(((t^2)/2)-1)+(t/2)-1, where t is the time in hours and y is the number of km's (measured in hundreds) the colony is away from the earth.

1.) What is the closest the colony will be to the earth and when will it happen? Round to three decimal places and label!

2.) Find (dy/dt)

3.) Where is (dy/dt)=0 on [2,5]?

4.) Using the graph of y and your soluctions from part 3, describe what the solutions in part 3 tell you about the graph of the function.

 

[Gene]

Need some info. Are you using a graphing calculator for this? Without one I don't see how 1) can be done. Usually I would expect 2) to be done first but I also don't see how to solve dy/dx=0 by algebra. Can you clarify what is going on in the class?

 

[Gene]

Since it is starting at t=pi/2 and 100 km the equation I would use for the distance from earth in hundreds of km is
1+sin(x²/2-1)+t/2-1 =
sin(x²/2-1)+t/2
If you don't have a graphing calculator you can go to
http://www.hostsrv.com/webmab/app1/MSP/ ... s&s3=basic
and type in
sin[x^2/2-1]+x/2
x=0 to 5
y=-2 to 4
to see what it looks like.
1) It will be closest at t=3.335 and y=.679 hundred km. That is about 40 mi high, 10 mi above the mesosphere where comets burn up. I think they will survive.
2) The derivitive of
sin(u)=cos(u)*du
u=t²/2-1
du=2t/2=t so
dy/dt=cos(t²/1-1)*(t)+1/2
3) There are 4 places (peaks and valleys) where it is zero. The graph will give the values.