Calculus II Take-Home Problems For Exam I--Due Date May 27 at the Beginning of Class

1.  Find the center of mass of the planar lamina whose boundaries are given by y = x² and y = x + 2 with constant density 2 units per square unit.  Click here to see a picture.  DPGraph Picture

2.  A rectangular box shaped container of water has dimensions 10' by 2' by 10' (see picture).  It is filled with water to a depth of 4 feet (again see picture).  Using a large tube the water is to be pumped out of the box.  In doing so all the water will need to be pumped to a height that is 4 feet above the top of the box after which it will fall to the ground.  Compute the total work done against gravity in pumping all the water to a height 4 feet above the top of the box.  Click on either picture below to see an animation of what is happening.

3.  A horse watering trough is filled with water to a depth of 2 feet.  The trough is in the shape of a supported hemisphere with a diameter across the top of 6 feet.  (See a picture below on the left.)  How much work against gravity is required to pump all the water to the top of the trough?  The density of water is 62.4 pounds per cubic foot.  Sketch the picture, model the problem in terms of a series, and give and compute the integral.

4.  A plate in the shape of a right triangle with legs of length 2 feet and 3 feet is submerged in a pool of water.  The 2 foot long leg is resting on the bottom of the pool parallel to the surface of the water.  The plate is vertical and the pool is 10 feet deep where the plate is.  (See a picture above on the right.)  What is the total fluid force of the water on one side of the plate?  Sketch the picture, model the problem in a series, give the integral and evaluate the integral.

5.  A pot of warm liquid has a temperature of 90⁰F.  It is moved into a room where the temperature is 60⁰F.  Ten minutes after it is moved the pot of liquid has a temperature of 80⁰F.  What will be the temperature of the pot of liquid to the nearest tenth of a degree one hour after it is moved into the 60⁰F room?

6.  Solve the differential equation below.

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        Lane Vosbury, Math Chair, Seminole Community College   email:  vosburyl@scc-fl.edu

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