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    Determining formula values.

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    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:
    Show work in this space.

    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:
    Show work in this space.

    c) Graph this function.
    Show graph here.

    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:
    Show work in this space. Use ^ to indicate the power.

    b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
    Answer:
    Show work in this space .

    c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
    Answer:
    Show work in this space.

    d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.
    Answer:
    Show work in this space.

    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:
    Show work in this space

    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:
    Show work in this space

    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:

    Show work in this space.

    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.

    Show work 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.
    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:
    Show work in this space.

    See attached file for full problem description.

    © BrainMass Inc. brainmass.com December 15, 2022, 5:44 pm ad1c9bdddf
    https://brainmass.com/math/basic-algebra/determining-formula-values-101331

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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 ...

    $2.49

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