Nonlinear Functions

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.

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b) Graph this function and show the graph over the valid range of the
variable x..

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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 121 cubic centimeters.


a) Write h as a function of r. Keep "(" in the function's equation.

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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.

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c) Graph this function.

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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 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).

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b) Calculate the return (A) if the bank compounds quarterly (n = 4).
Round your answer to the hundredth's place.

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

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

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

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f) If a bank compounds continuous, then the formula becomes simpler,
that is

where e is a constant and equals approximately 2.7183. Calculate A with
continuous compounding. Round your answer to the hundredth's place.

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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.

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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.

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, 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.

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b) FORMTEXT Show graph here .

5) Logarithms:

a) Using a calculator, find log 1000 where log means log to the base of
10.

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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:



. Round your answer to the hundredth's place.

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