Math 126-104
Spring 2013
Quiz 8, Feb. 15
Carter
Name
A bag of sand originally weighing 144 lb was lifted at a constant rate. As it rose, sand also
leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft.
How much work was done lifting the bag this far? (Neglect the weight of the bag and the lifting
equipment.)
Solution. Since the weight of the bag is a linear function of its height, we compute the changes
∆x = (x1 − x0 ) = 18 and ∆y = (y1 − y0 ) = −72 for (x0 , y0 ) = (0, 144) and = (x1 , y1 ) = (18, 72).
Thus the weight as a function of height is w = 144 − 4x. To compute the work, we use the integral
Z
W =
0
18
18
2
(144 − 4x) dx = 144x − 2x = 144 · 18 − 2 · 182
0
= 8 · 182 − 2 · 182 = 182 [8 − 2] = 6 · 324 = 36 · 54 = 2025 − 81 = 1944 ft.lbs.