math20c Meesue Yoo fall09 homework practice 14.8

# math20c Meesue Yoo fall09 homework practice 14.8 - p is a...

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Extra Homework on 14.7 and 14.8 Section 14.7 1. Find the points on the plane x - y + z = 4 that is closest to the point (1 , 2 , 3). 2. Find three positive numbers whose sum is 100 and whose product is a maximum. 3. Find the volume of the largest rectangular box in the ﬁrst octant with three faces in the coordinate planes and one vertex in the plane x + 2 y + 3 z = 6. 4. Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9 x 2 + 36 y 2 + 4 z 2 = 36. Section 14.8 1. Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter
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Unformatted text preview: p is a square. 2. Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. (Use Heron’s formula for the area A = p s ( s-x )( s-y )( s-z ) , where s = p/ 2, and x , y , z are the lengths of the sides). 3. Find the points on the sphere x 2 + y 2 + z 2 = 4 that are closest to and farthest from the point (3 , 1 ,-1). 4. The plane x + y + 2 z = 2 intersects the paraboloid z = x 2 + y 2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin....
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