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    Linear and Non-Linear Equations : Finding Minimum or Maximum Value

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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.
    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.
    Answer:
    Show work in this space.

    b) What is the measurement of the height if the radius of the cylinder is 3 centimeters?
    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).
    Answer:
    Show work in this space .

    c) Calculate the return (A) if the bank compounds monthly (n = 12).
    Answer:
    Show work in this space.

    d) Calculate the return (A) if the bank compounds daily (n = 365).
    Answer:
    Show work in this space.

    e) What observation can you make about the 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.
    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).
    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?
    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.

    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:

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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:

    The area of the base is: (4-2x)*(6-2x)
    The height is: x, so the volume V = (4-2x)*(6-2x) *x

    b). Graph this function.
    You can draw it according to the function. Pay attention that two points (2, 0) (0, 0). i.e, when
    x = 2, 0, the volume is 0 and also x , V > 0, i.e. x cannot be negative, and the volume cannot be negative. The range of x is 0<x< 2, because (4-2x) > 0

    c). From the graph, we can see that at the point where V is maximum, then the tangent of point(x,V) is 0, i.e, dV/dx = 0 at point (x, V)

    dV/dx = d((4-2x)*(6-2x) *x))/dx = ...

    Solution Summary

    Equations are written from word problems and solved for optimum value.

    $2.19