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Lesson 4 - Quadratic Applications

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Lesson 4 - Quadratic Applications Powered By Docstoc
					Quadratic Functions…
  and their
applications!
  For a typical basketball shot, the ball’s
height (in feet) will be a function of time
in flight (in seconds), modeled by an
equation such as h = -16t2 +40 t +6.
 a) What is the maximum height of the ball?

  b) When will the shot reach the height of
 the basket? (10 feet)
 c) When will the ball hit the floor, if it
 missed the basket entirely?
   a) What is the maximum
      height of the ball?
 Put it in your calculator!
 Use your zooms and change   your
  window until you see the maximum.
 Find the maximum!


    Answer: The maximum
  height of the ball is 31 feet!
  b) When will the shot reach the
  height of the basket? (10 feet)
 Key    words to highlight:
 When (so we are looking for our x)
 Height of the basket (10 feet)

 Put   10 in for y2 and find the…
          INTERSECTION!

   Answer: 2.4 seconds!
c) When will the ball hit the floor, if
   it missed the basket entirely?
 What   do we put in for y2?
   y2 = 0
 Now    find the intersection!

  Answer: The ball will hit the
   floor after 2.64 seconds!
                   YOU DO:
 The height, H metres, of a rocket t
 seconds after it is fired vertically upwards
 is given by H (t )  80 t  50 t 2 , t  0
     How long does it take for the rocket to reach
      its maximum height?
     What is the maximum height reached by the
      rocket?
     How long does it take for the rocket to fall
      back to earth?
       Mrs. Holst (who loves to swim!) is putting in a swimming
pool next to her house. She wants to put a nice, rectangular
privacy fence around it, but she can only afford to pay for 50 feet
of fencing. If she does not need a fence on the part adjacent to her
house, what are the dimensions of the fence with the largest area
she could have for her pool?
                           Help me get the most
   My house!
                           space for my money!
                                   2x + y = 50
                                   y = 50 - 2x
                                   Area = x 50 – 2x
                                            y
                                      A = x(50 – 2x)
                                      A = 50x – 2x2
                                      Now graph it!
                              x ft.

                           x ft.               y ft.
My pool will
 go here!      My future
                fence!
                 Maximum Area                Put it in your
                                            calculator and
       350
       300                                     find the
       250                                     what???
                                              MAXIMUM
Area




       200
                                              Do we need the
       150
                                              x value or the y
       100
                                                  value?
        50
         0                                      x value!
                                                  x = 12.5 ft.
             0      10            20   30
                                             thus y = 50 – 2(12.5)
                         Length                     y = 25
                                            Dimensions of the
                                                 Fence:
                                              25 ft x 12.5 ft
    A farmer wants to build two
rectangular pens of the same size
next to a river so they are separated
by one fence. If she has 240 meters
of fencing and does not fence the
side next to the river, what are the
dimensions of the largest area
enclosed? What is the largest area?
 Step 1: Draw a figure!


xm      xm      xm




         ym
   Step 2: Set up your equations!
Perimeter equation        3x + y = 240
Area equation                A = xy
Solve for y!               y = 240 – 3x
Substitute y into the
area equation             A = x(240 – 3x)
Distribute the x.          A = 240x – 3x2
               Now what type of function do we have????

                                                       So
                                                    graph it!
               Step 3: Graph it!
Remember: There are two questions in the problem.
        1. What are the dimensions of the largest area
        enclosed?
        2. What is the largest area?
So when we graph and find the maximum, are we looking for the x or y
for number 1?
                        x!
So when we graph and find the maximum, are we looking for the x or y
for number 2?
                        y!
The Chesapeake Bay
       Average Monthly Temperatures of
             the Chesapeake Bay
Month   Jan   Feb Mar    Apr May Jun    Jul   Aug Sep   Oct   Nov Dec
Temp    31    34    44   54   64   72   76    75   68   57    47   36

        1. Turn on your STAT PLOT and Diagnostics (2nd 0 x-1)
        2. Enter your data in L1 and L2
        3. Look at the data you have entered. What is the
        temperature doing? Now let’s actually look at the STAT
        PLOT (Zoom 9).
        4. Which function that we’ve studied would best model
        the data?
                   Do a quadratic regression!
                              STAT CALC 5
 What is the r2 value?
        r2 = .927
  This tells us that 92.7%
of the time, the model is
a good predictor, and the
closer this value is to 1,
the closer the data is to
the model.
              Analysis
          to the model, what month does
 According
 the maximum temperature occur?

               June!
          to the model, during what
 According
 months would the temperature be 50°?

  March and October
   Darryl is standing on top of the bleachers and
  throws a football across the field. The data that
follows gives the height of the ball in feet versus the
         seconds since the ball was thrown.
Time
         0.2 0.6 1 1.2 1.5 2 2.5 2.8 3.4 3.8 4.5
Ht.       92 110 130 134 142 144 140 132 112 90                                   44
      a. Show a scatter plot of the data. What is the independent variable, and
         what is the dependent variable?
      b. What prediction equation (mathematical model) describes this data?
      c. When will the ball be at a height of 150 feet?
      d. When will the ball be at a height of 100 feet?
      e. At what times will the ball be at a height greater than 100 feet?
      f. When will the ball be at a height of 40 feet?
      g. When will the ball hit the ground?
a. Show a scatter plot of the data. What is
the independent variable, and what is the
          dependent variable?

  Independent variable (x): Time! (always!)
  Dependent variable (y):    Height
  b. What prediction equation
(mathematical model) describes
          this data?


QUADRATIC!!
c. When will the ball be at a height
          of 150 feet?
 Height (y)
 Put 150 in y2.

         What happened?!? Explain.
d. When will the ball be at a height
          of 100 feet?

   Put 100 in y2 and find the intersection!


               .34 seconds
                    and
               3.65 seconds
e. At what times will the ball be at
 a height greater than 100 feet?


    .34  x  3.65
f. When will the ball be at a height
           of 40 feet?



     4.53 seconds
 g. When will the ball hit the
        ground?

Put 0 in y2 and find the intersection!



 4.98 seconds
Now try it on
 your own!

				
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