Calc Word problems Help! Lost :(

[jkitt]

*A rectangle with sides parallel to to the coordinate axes is inscribed in the ellipse

(4x^2)+(9y^2)=36

Find the dimensions of the rectangle of greatest area.


*The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. If two light sources are 40 feet apart and their intensities are 40 and 70 respectively, at what point between them will the sum of their illuminations be a minimum?


Solution:

Let x be the distance at which the sum of the illuminations be minimum. Then x=_____ feet.

*A racer can cycle around a circular loop at the rate of 6 revolutions per hour. Another cyclist can cycle the same loop at the rate of 10 revolutions per hour. If they start at the same time (t=0), at what first time are they farthest apart?

PLZ Help/explain as much as possible I'm so-o lost.

 

[Gene]

I am assuming you can take a derivitive and just need help with the equations..

Draw the elipse centered on P(0,0). Draw the rectangle. You know
The area of the rectangle is A = 4 xy.
(4x^2)+(9y^2)=36
x = sqrt(36-9y^2)
A = 4sqrt(36-9y^2)y
Find dA/dy
Set it equal to 0
Solve for y
Solve for x in x = sqrt(36-9y^2)
The dementions are 2x by 2y

The illumination is
I₁/x² + I₂/d₂²
d₂ = 40 - x
Substitute in previous equation and solve for dI/dx

The distance between them is could be across the circle or around the track. The answer for t hours would be the same. That is when they are on opposite sides of the circle. Using the second equation:
The first rider goes 2*pi*6 radians per hour.
The second goes 2*pi*10 radians per hour.
The difference is 2*pi*4 radians per hour.
So 2*pi*4*t = pi
If you feel a need to use calculus let us know but since it isn't necessary in solving this one...