This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATH 271 Test #3T Solutions (1) (20 points) A tank in the shape of a sphere with a radius of 5 meters is filled halfway with water (density: 1000 kg/m 3 ). The center of the tank is 20 meters underground. How much work does it take to pump the water to the surface? Evaluate the integral. (Note g = 9 . 8 m/s 2 .) Solution: The amount of work needed to pump the water to the surface is the integral of the weight of the water (in a thin slice) times the distance it has to be moved. The weight of the water is the density of water times g , times the crosssection area at a given depth, times dx . If the depth of the water is x , then the crosssection area of the tank is r 2 , where r 2 + ( x 20) 2 = 5 2 (because the tank is 20 meters below the surface and is spherical with radius 5 ). The area is only counted for the bottom half of the tank, so 20 x 25 . The integral which represents the work needed to pump out all the water is 25 Z 20 (1000)(9 . 8)( r 2 ) xdx = 9800 25 Z 20 (5 2 ( x 20) 2 ) xdx. To evaluate the integral, expand ( x 20) 2 = x 2 40 x +400 and multiply out the polynomial to get 9800 25 Z 20 25 x x 3 + 40 x 2 400 x = 9800  x 4 4 + 40 x 3 375 2 x 2 25 20 = 56 , 123 , 243 . 79 J The substitution u = x 20 could also be made. Grading: This problem was graded on a 5 10 15 20 basis. Grading for common mistakes: 5 points if the integral wasnt evaluated; +7 points (total) if the mass times a depth of 20 m or 25 m was used; +10 points (or more) if the expression involved an integra; +15 points (or more) if the integral of an area was found. 1 (2) Consider the two circles shown in the diagram below, which have the equations ( x 2) 2 + y 2 = 4 2 and ( x +2) 2 + y 2 = 4 2 . The region inside both circles is sometimes called a vescica piscis ....
View
Full
Document
This note was uploaded on 02/24/2009 for the course MAT 26996 taught by Professor Hurlbert during the Spring '08 term at ASU.
 Spring '08
 Hurlbert
 Math, Calculus

Click to edit the document details