math121A winter 2010 practice hw3

math121A winter 2010 practice hw3 - Math 121A, HW3 (1)...

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Unformatted text preview: Math 121A, HW3 (1) Summation Formula: We have used the formula n i= i=1 n(n + 1) 2 on several occasions. Prove this formula. (2) Cylindrical Tank Problem: A cylindrical tank (oriented vertically) is full of water. A drain in the bottom of the tank is opened. You want to know: (a) The time it takes for the tank to empty completely. (b) The amount of water remaining in the tank at any given time from the moment the drain is opened. Use the following data to help you answer the above questions: Time (min) Water remaining (gal) 0 600 1/2 14161/24 1 3481/6 3/2 4563/8 2 1682/3 5/2 13225/24 3 1083/2 7/2 12769/24 4 1568/3 9/2 4107/8 (3) Expanding Rectangle Problem: A rectangle ABCD starts with side lengths of 7 and 15, and all of the sides of the rectangle are getting longer at the same constant rate (so the rectangle is maintaining a rectangular shape). Let m and n be the lengths of the sides of the rectangle at a particular time, and consider two increments. The first is when one side length is between m − h/2 and m + h/2, while the other side length is between n − h/2 and n + h/2. The second is when one side length is between m and m + h, while the other is between n and n + h. (a) What are the rates of change of area with respect to time over these two increments? (b) What about the rates of change of the perimeter with respect to time over these increments? (c) What about the rates of change of the length of the diagonal with respect to time over these increments? 1 ...
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