MAT1322D Solution to Final Examination December 2017 2
3. Suppose a pool has the shape of an inverted truncated pyramid. The top of the pool at the ground level is a square with side length 20 meters, and the bottom of the pool is a square with side length 10 meters. The depth of the pool is 5 meters. The pool is filled with water with density kg / m3. Let xbe the distance between a horizontal layer of water and the top of the pool. Denote the acceleration of gravity be gm / sec2. Then the work, in Joules, needed to pump the water in the pool to a point 2 meters above the ground is calculated by the integral
Solution. (D) A horizontal layer of water in the pool is a square with side-length L(x) = 10 + 2 (5 –x) = 20 –2x. The volume of this layer with thickness dxis V(x) = (L(x))2= (20 –2x)2dx. The weight of this layer of water is w(x) = gV(x) = g(20 –2x)2dx. The work needed to pump this layer of water to a point 2 meters above the ground is W(x) = w(x)(x+ 2) = g(20 –2x)2(x+ 2)dx. The total work is W= 520(202 ) (2)gxxdx.