# sol3 - Solutions to Final Exam Math 20C Dec 9 2008(1 The...

This preview shows pages 1–2. Sign up to view the full content.

Solutions to Final Exam, Math 20C, Dec 9, 2008 (1) The formula for the volume of a cone is not needed for this problem. Piece A has the same shape as the big cone, but all three of its dimensions are reduced by a factor of 4/5, as we can see using similar triangles. Thus the volume of piece A is (4/5)^3 times the volume of the big cone. Since (4/5)^3 = 64/125 > 1/2, piece A has more than half the volume of the big cone, so piece A has greater volume than piece B (but not by much). (2) The inner integral is the integral of 1, from z = - sqrt(1 - x^2 - y^2) to z = sqrt(1 - x^2 - y^2). Thus the inner integral equals 2sqrt(1 - x^2 - y^2). It follows that the volume of the unit sphere is the double integral of 2sqrt(1 - x^2 - y^2) dx dy over the unit disk on the floor centered at the origin. Changing to polar coordinates, this becomes the double integral of 2sqrt(1-r^2) r dr dt (where t = theta). The inner integral goes from r=0 to r=1, and the outer integral goes from t=0 to t=2Pi.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern