1. An open-top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.
   1. Find the function V that represents the volume of the box in terms of x .
      
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   2. Graph this function.
      
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   3. Using the graph, what is the value of x that will produce the maximum volume?
      
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   4. Write h as a function of r .
      
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   5. What is the measurement of the height if the radius of the cylinder is 2 centimeters?
      
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   6. Graph this function.
      
      
2. The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of Deposit) is given by
   \[A=P{{\left( 1+\frac{r}{n} \right)}^{nt}}\]
   A is the amount of returned.
   P is the principal amount initially deposited.
   r is the annual interest rate (expressed as a decimal).
   n is the compound period.
   t is the number of years.
   
   Suppose you deposit $10,000 for 2 years at a rate of 10%.
   1. Calculate the return ( A ) if the bank compounds annually ( n = 1).
      
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   2. Calculate the return ( A ) if the bank compounds quarterly ( n = 4).
      
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   3. Calculate the return ( A ) if the bank compounds monthly ( n = 12).
      
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   4. Calculate the return ( A ) if the bank compounds daily ( n = 365).
      
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   5. What observation can you make about the increase in your return as your compounding increases more frequently?
      
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   6. If a bank compounds continuous, then the formula takes a simpler form, that is \[A=P{{e}^{rt}}\] where e is a constant and equals approximately 2.7183.
      Calculate A with continuous compounding.
      
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   7. Now suppose, instead of knowing t , we know that the bank returned to us $15,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t ).
      
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   8. A commonly asked question is, "How long will it take to double my money?" At 10% interest rate and continuous compounding, what is the answer?
      
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3. For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, \[A=P{{\left( 1+\frac{r}{n} \right)}^{nt}}\] , let r = 10%, P = 1, and n = 1 and give the coordinates ( t , A ) for the points where t = 0, 1, 2, 3, 4.
   1. Show coordinates in this space: The coordinates are
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4. Logarithms:
   1. Using a calculator, find log 10000 where log means log to the base of 10.
      
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   2. Most calculators have 2 different logs on them: log, which is based 10, and ln, which is based e . In computer science, digital computers are based on the binary numbering system which means that there are only 2 numbers available to the computer, 0 and 1. When a computer scientist needs a logarithm, he/she needs a log to base 2 which is not on any calculator. To find the log of a number to any base, we can use a conversion formula as shown here:
      \[{{\log }_{b}}a=\frac{\log a}{\log b}\]
      Using this formula, find \[{{\log }_{2}}10000\] .

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