Volume, Interest and Logarithm Questions

1) An open-top box is to be constructed from a 4 foot by 6 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 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 = πr2h 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. Keep "π" in the function's equation.

b) What is the measurement of the height if the radius of the cylinder is 3 centimeters? Round your answer to the hundredth's place.

c) Graph this function.

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

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

Carry all calculations to 6 decimals on all assignments then round the answer to the
nearest cent.

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). Round your answer to the hundredth's place.

c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.

d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.

e) What observation can you make about the size of 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. Round your answer to the hundredth's place.

g) Now suppose, instead of knowing t, we know that the bank returned to us $25,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth's place.

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? Round your answer to the hundredth's place.

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. Round your answer to the hundredth's place.

a) Show coordinates in this space.

b) Show graph here.

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 ln, 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 . Round your answer to the hundredth's place.