Algebra: Analysis and Graphing

1) An open top box is to be constructed from a 4 by 6 rectangular cardboard by cutting out equal squares at each corner and then 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 and show the graph over the valid range of the variable x.
c) Using the graph, what is the value of x that will produce the maximum volume?

2) The volume of a cylinder (think about the volume of a can) is given by V= pi r squared h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 121 cubic centimeters.

a) Write h as a function of r.
b) What is the measurement for the height if the radius of the cylinder is 3 centimeters?
c) Graph this function.

3) The formula for calculating the amount of money returned for deposit money into a bank account of CD is given by the following: A= P {1 + r/n} nt

A is the amount returned
P is the principal amount deposited
r is the annual interest rate (decimal)
n is the compound period
t is the number of years

Suppose you deposit $20,000 for 3 years at a rate of 8%.

a) Calculate the return A if the bank compounds annually (n=1).
b) Calculate the return A if the bank compounds quarterly (n-4), and carry all calculations to 7 significant figures.
c) Calculate the return A if the bank compounds monthly (n-12), and carry all calculations to 7 significant figures.
d) Calculate the return A if the bank compounds daily (n-365), and carry all calculations to 7 significant figures.
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 becomes simpler, that is A=Pe^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 has returned to us $25,000 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 8% 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 {1+ r/n} nt,
let r= 8%, 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.

5) Logarithms:
a) Using a calculator, find log 1000 where log means log to the base of 10.
b) Most calculators have 2 different logs on them: log, which is base 10, and 1n, which is base 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 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 = log a/ log b. Using this formula, find log 2 1000.