# Determining formula values.

1) An open-top box is to be constructed from a 4 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.

Answer:

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

Show Graph here.

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

Answer.

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.

Answer:

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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.

Answer:

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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

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).

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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.

Answer:

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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.

Answer:

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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.

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 = 8%, P = 1, and n = 1 and give the coordinates (t,A) for the points where t = 0, 1, 2, 3, 4. Round the A value to the tenth's place.

a) Show coordinates in this space.

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b) Show graph here.

5) Logarithms:

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

Answer:

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:

Using this formula, find . Round your answer to the hundredth's place.

Answer:

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See attached file for full problem description.

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#### Solution Preview

1) An open-top box is to be constructed from a 4 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.

Answer:

You are cutting an x out of each side so the dimensions of the bottom of the box are (6-2x) and (4-2x). Imagine folding up the corners, that will be the height of the box, x

The volume of the box is l*w*h = (6-2x)(4-2x)x

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

Show Graph here.

Keeping in mind x can't be greater than 2, otherwise the width of the box would be less than 0, the graph is shown below:

See attached

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

Answer. Using the graph, to find the max.. only between 0 and 2, we get x = .78 ft

2) The volume of a cylinder (think about the volume of a can) is given by V = πr^2h 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 ...