# Polynomial, Rational, Exponential, and Logarithmic Functions

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.

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b) How do I Graph this function?

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c) Using the graph, how do I find 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.

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b) How do I find out what the measurement of the height is if the radius of the cylinder is 2 centimeters?

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c) How do I Graph this function?

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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 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) How do I calculate the return (A) if the bank compounds annually (n = 1).

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b) Please show me how to calculate the return (A) if the bank compounds quarterly (n = 4).

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c) How do I calculate the return (A) if the bank compounds monthly (n = 12).

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d) How do I calculate the return (A) if the bank compounds daily (n = 365).

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

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

where e is a constant and equals approximately 2.7183.

Calculate A with continuous compounding.

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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).

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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?

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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, , let r = 10%, P = 1, and n = 1 and give the coordinates (t, A) for the points where t = 0, 1, 2, 3, 4.

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5) Logarithms:

a) Using a calculator, how do I find log 10000 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 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:

Using this formula, how do I find ?

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6) The loudness of sound is based on intensity level measured in decibels using a logarithmic scale and is relative to (a ratio of) the weakest sound the ear can hear.

How do I figure out how sound is measured? I need to include the following items:

? The formula for measuring sound.

? Pick a specific sound, give the decibels of the sound, and explain what this measurement means.

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