Calculus III optimization??

Question

1) A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x+2y = 2. Find the maximum area of this traingle.

2) A rectangular box is to be inscribed inside the ellipsoid 2x^2 +y^2+4z^2 = 12. Find the largest possible volume for the box.


How the hell am I supposed to solve these questions using partial derivatives.?

Answer 1

The first problem is not stated correctly. You say that it is a rectangle -- but then ask for the area of a triangle. I assume you want it to be a rectangle.

You don't need partial derivatives to solve this one. The area of the rectangle will be X*Y -- where both are points on the line segment.

Since X+2Y=2, you get

Y = 1-X/2

This means that the area is X*(1-X/2) = X-X^2/2

Take the first derivative of this and set it equal to zero.

For the second equation, you want to maximize V(x,y,z) = x*y*z subject to the constraint that 2x^2 +y^2+4z^2 = 12.

You can do this using Lagrange Multipliers -- which involves taking partial derivatives.

Answer 2

Part 1.) Should be easy. You know that the area of the rectangle is the product of x*y. You also know that x and y are related by the equation: x+2y=2. Rearranging this, you should see that y=(2-x)/2. Thus the area (A) is a function of x: A = x*(2-x)/2.
Differentiate A with respect to x, find the point where the derivative is zero, and that should be the point (x-value) of greatest area. You can find y, from x.

Part 2.) Is a bit harder, but the principles should be the same. You may need to work on a piece of the ellipsoid, rather than the whole, say just the section where all axes are positive. Then, find the function for z as a dependent of x and y: z=f(x,y).
After that, state a function for the volume of the box bounded by the axes, and that piece of the ellipsoid. Also find the total (?) differential of the Volume w.r.t. x and y. Set that to 0, then come up with a final equation which relates the max area to a function of x and y only. Differentiate y w.r.t. x, set the derivative to zero and find your x-point. Then back fill to find the y- and z-points.

I haven't tried this in a very long time, but believe that is how it can be done.

Answer 3

To answer your last question first : read your text...not just look at the words. If you still don't understand see your instructor with specific questions.
1) the area is xy where (x,y) is on the given line. YOU should be able to write the area now as a function of x.
2) If (x.y.z) is the point where the box touches the ellipsoid then the volume is 8xyz, solve the ellipsoid equation for z ,put it into the volume equation, and use the theorem on max/min for functions of two variables. Yes it does involve partial derivatives.