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.

a) Find the function V that represents the volume of the box in terms of x.

b) Graph this function.

c) Using the graph, what is the value of x that will produce the maximum volume?

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2) The volume of a cylinder (think about the volume of a can) is given by V = πr²h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 100 cubic centimeters.

a) Write h as a function of r.

b) What is the measurement of the height if the radius of the cylinder is 2 centimeters?

c) Graph this function.

3) 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%.

a) Calculate the return (A) if the bank compounds annually (n = 1).

b) Calculate the return (A) if the bank compounds quarterly (n = 4).

c) Calculate the return (A) if the bank compounds monthly (n = 12).

d) Calculate the return (A) if the bank compounds daily (n = 365).

e) What observation can you make about the increase in your return as your compounding increases more frequently?

f) 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.

g) 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).

h) 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?

4) 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.

a) Show coordinates in this space:

b) Show graph here

5) Logarithms:

a) Using a calculator, find log 10000 where log means log to the base of 10.

b) 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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