2.(a) Find, in spherical coordinates, the equation of the 3-dimensional curve formed by the cone j = p /4 and the plane

x + y + z = 1.

(b) Find the point(s) on the curve in (a) that is (are) most distant from the origin.

x = r cos(q )sin(j ),y = r sin(q )sin(j ), z = r cos(j )

in the equation x + y + z = 1 we get

r cos(q )sin(j ) + r sin(q )sin(j ) + r cos(j )=1.

Next, since j = Pi/4, sin(j ) = cos(j ) = 1/Ö 2 it follows that

r cos(q )+r sin(q )+r = Ö 2

and the equation of the curve is

b) Since r is also the distance from the point on the curve to the origin, r will be maximal when

g(q ) = 1 +cos(q )+sin(q )

is minimal. Taking a derivative to find the min and max we find that

g'(q )= -sin(q )+cos(q )= 0 Þ q = Pi/4 or q = 5Pi/4

Checking by computation we find that the minimum, 1 - Ö 2 occurs when q = 5Pi/4. Thus the maximal distance is