Math Lab Module 3

For each of the following problems, show all work.  Simplify and clearly indicate all answers.

1.  A ball is thrown vertically upward from the top of a building 160 feet tall with an initial

     velocity of 48 feet per second.  The distance d (in feet) of the ball from the ground after

     t seconds is d(t) = 160 + 48t – 16t². 

     a.  After how many seconds does the ball strike the ground?  Write your answer in a complete sentence with proper grammar and correct spelling.

     b.  When will the ball reach its maximum height?  What is the maximum height of the ball?  Write your answers in a complete sentence with proper grammar and correct spelling.

2.  Laura owns and operates Aunt Linda’s Pecan Pies.  She has learned that her profits, P(x), from the sale of x cases of pies, are given by P(x) = 150x – x².

     a.  The company will “break-even” when the profit is zero.  How many cases of pies should Laura sell in order to break-even? (Solve for x when P(x) = 0.)  Write your answer in a complete sentence with proper grammar and correct spelling.

     b.  How many casesof pies should she sell in order to maximize profit?  What is the maximum profit?  Write your answer in a complete sentence with proper grammar and correct spelling.

Math 1314                                                     Lab Module 3                                                           page 3

3.  For f(x) = x³ – 7x² + 8x + 16, 

    a.  Find f(10) using synthetic division.            

    b.  Is 10 a zero of the function?  Explain in a complete sentence with proper grammar and correct spelling.            

    c.  Use synthetic division to determine if x + 1 is a factor of f(x).                 

    d.  Is –1 a zero of the function?  Explain in a complete sentence with proper grammar and correct spelling.         

    e.  List all of the zeros and their multiplicities of the polynomial.

    f.  Write the polynomial function as a product of linear factors.

Math 1314                                                     Lab Module 3                                                           page 4

4.  For the function 

1. State the degree of the polynomial.

• State thenumber of zeros the polynomial function will have.             

    c.  Use the Rational Zero Theorem to list all of the possible rational zeros

    d.  Use your calculator to determine which numbers in the list of rational zeros are probable rational zeros.                                                           

    e.  Use synthetic division to verify one rational zero. 

    f.  Use synthetic division or other algebraic methods to find all remaining zeros.  List all of the zeros of the polynomial function.

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5.  Analyze and sketch the polynomial functions and complete the charts below. State the degree and sign of the leading coefficient of the polynomial functions.  Determine the end behavior of the graph of the functions.  For 5b, write the polynomial function as a product of linear factors (in factored form).

     a.  f(x) = –7(x + 2)³(x – 1)²(x – 3)  

Zeros   Multiplicity   Crosses/Touches

       Degree

        Sign

        End behavior                                                                                                       

      b.  m(x) = x³ – 4x²                       

      Factored form:  m(x) =

      Degree                                          Zeros   Multiplicity   Crosses/Touches

       Sign

       End behavior

6.  Based on data from the U.S. Department of Agriculture, the average number of acres per farm x years after 2000 can be approximated by the model below.  (Round answers to 2 decimal places.)

     a.  Use the model to estimate the average number of acres per farm in 2005.

     b.  Use the model to predict the average number of acres per farm in 2012.

     c.  Find and interpret the zero of the rational function.  Does this result make sense within the

          context of the problem?Answer in complete sentences using proper grammar and correct spelling.

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7.  A rare species of insect was discovered in the rain forest.  In order to protect the species,   environmentalists declare the insect endangered and transplant the insects into a protected area.  The population of the insect t months after being transplanted is given by P(t).

             P(t) =

    a.  How many insects were discovered?  In other words, what was the population when t = 0?

    b.  What will the population be after 5 years? Round to the nearest whole insect.        

    c.  Determine the horizontal asymptote of P(t).    

         Describe what the horizontal asymptote means in the context of the problem.  Use the value of

         the horizontal asymptote in the explanation.  Answer in a complete sentence using proper grammar and correct spelling.

    d.  Sketch the graph of P(t).

8.  State the domain, vertical asymptote and slant asymptote of the function C(x) = .

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