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### Section 6-6 : Work

1. A force of $$F\left( x \right) = x{{\bf{e}}^{ - 2{x^{\,2}}}} + 6x - 2$$ acts on an object. What is the work required to move the object from $$x = 1$$ to $$x = 4$$?
2. A force of $$F\left( x \right) = 4\cos \left( {2x} \right) - 7\sin \left( \frac{1}{2}x} \right$$, $$x$$ is in meters, acts on an object. What is the work required to move the object 10 meters to the right of $$x = 2$$?
3. A force of $$F\left( x \right) = \sin \left( x \right){{\bf{e}}^{\cos \left( x \right)}} - 4x + 1$$, $$x$$ is in meters, acts on an object. What is the work required to move the object 6.5 meters to the left of $$x = 9$$?
4. A spring has a natural length of 25 cm and a force of 3.5 N is required to stretch and hold the spring to a length of 32 cm. What is the work required to stretch the spring from a length of 30 cm to a length of 45 cm?
5. A spring has a natural length of 9 inches and a force of 7 lbs is required to stretch and hold the spring to a length of 21 inches. What is the work required to stretch the spring from a length of 12 inches to a length of 30 inches?
6. A cable with mass 2 kg/meter is lifting a load of 50 kg that is initially at the bottom of a 75 meter shaft. How much work is required to lift the load 40 meters?
7. A cable with mass 1.5 kg/meter is attached to a bucket that has mass 75 kg. Initially there is 500 kg of grain in the bucket and as the bucket is raised 2 kg of grain leaks out of a hole in the bucket for every meter the bucket is raised. The bucket is 200 meters below a bridge. How much work is required to raise the bucket to the top of the bridge?
8. A tank of water is in the shape of a cylinder of height 25 meters and radius of 7 meters. If the tank is completely filled with water how much work is required to pump all of the water to the top of the tank. Assume that the density of water is 1000 kg/m3.
9. A tank of water is in the shape of an inverted pyramid that is 18 feet tall and whose top is a square with sides 4 feet long. If there is initially 12 feet of water in the tank determine the amount of work needed to pump all of the water to the top of the tank. Assume that the density of water is 62 lb/ft3.
10. A tank of is the shape of the lower half of a sphere of radius 6 meters. If the initial depth of the water is 4 meters how much work is required to pump all the water to the top of the tank. Assume that the density of water is 1000 kg/m3.