MATH 275 – Section 001 – Quiz 7
You may work with other class members on this quiz, but you may not receive
assistance from people not in MATH 275 (Section 001). You must show all of
your work to receive full credit. Do all your work on other sheets of paper and
be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’
ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your
work on this quiz must be handed in by Monday, 21 October 2002 at 9:40 a.m.
GOOD LUCK!
1) Consider a triangle whose angles are A,B, and C. Use Lagrange multipiers
to prove that the expression
sin Asin Bsin C(1)
is maximized when the triangle is equilateral. Provide an argument that explains
why your result maximizes (and not minimizes) (1).
2) Use Lagrange multipliers to find the point closest to the origin on the inter-
section of the planes y+ 2z= 12 and x+z= 6. Provide an argument to explain
why the point you found represents the closest point and not some other type
of extreme point.
3) Let R={(x, y) : 0 ≤x≤4,0≤y≤6}. Compute the integral
Z
R
Z(2x+ 3) dA
in three different ways:
a) as an iterated integral, integrating first with respect to x, then with
respect to y;
b) as an iterated integral, integrating first with respect to y, then with
respect to x;
c) using geometrical arguments (i.e., do not compute any antiderivatives).
4) Find the volume of the solid bounded by:
•the x-yplane
•the y-zplane
•the x-zplane
•the plane x= 2
•the place y= 3
•the plane 3x+ 2y−z+ 6 = 0.