The equation for the surface of an ellipsoid is given by (x/a)^2 + (y/b)^2 + (z/

Question

The equation for the surface of an ellipsoid is given by (x/a)^2 + (y/b)^2 + (z/c)^2=1.

a) For a fixed value of x, find the maximum value of y on the ellipsoid surface, as a function of x.

    For fixed values of x and y, find the value of z on the ellipsoid surface, as a function of x and y.

b) Using the result of part a, show that the volume of the ellipsoid is (4/3)(Pi)abc.

c) Let M be the mass of the ellipsoid, and that mass is uniformly distributed over the ellipsoid volume.  Determine the moment of inertia with respect to the x-axis.

Explanation / Answer

1. x/a + y/b + z/c = 1 for a
x/a = 1- y/b - z/c
1/a = (1- y/b - z/c) / x
a = x / (1- y/b - z/c)

2. v= 2 (ab + bc + ca) for a
v/2 = ab + bc + ca
v/2 - bc = a (b + c)
a = (v - 2bc) / (2b + 2c)

3) a= 2piR^2 + 2piRh for positive r
GOOD LUCK ON THIS ONE :]

4) A= P + nrP for P
A = P(1 + nr)
A / (1-nr) = P

5) 2x-2yd= y + xd for d
2x = y + xd + 2yd
2x = y + d(x+ 2y)
2x - y =d(x+ 2y)
(2x - y) / (x+ 2y) = d

6) 2x/4pi + 1-x/2= 0 for x
2x/4pi - x/2 = -1
(2x - 2pix) / 4pi = -1
2x(1-pi) / 4pi = -1
2x(1-pi) = -4pi
x = -4pi / 2(1-pi)
x= -2pi/ (1-pi)

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