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Quadratics in Real Life

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Quadratics in Real Life Quadratic functions are more than algebraic curiosities they are widely used in science, business, and engineering. The U-shape of a ... – PowerPoint PPT presentation

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Title: Quadratics in Real Life


1
Quadratics in Real Life
2
  • Quadratic functions are more than algebraic
    curiositiesthey are widely used in science,
    business, and engineering.
  • The U-shape of a parabola can describe the
    trajectories of water jets in a fountain and a
    bouncing ball, or be incorporated into structures
    like the parabolic reflectors that form the base
    of satellite dishes and car headlights.
  • Quadratic functions help forecast business profit
    and loss, plot the course of moving objects, and
    assist in determining minimum and maximum values.
  • Most of the objects we use every day, from cars
    to clocks, would not exist if someone, somewhere
    hadn't applied quadratic functions to their
    design.

3
We commonly use quadratic equations in situations
where two things are multiplied together and they
both depend of the same variable. For example,
when working with area, if both dimensions are
written in terms of the same variable, we use a
quadratic equation. Because the quantity of a
product sold often depends on the price, we
sometimes use a quadratic equation to represent
revenue as a product of the price and the
quantity sold. Quadratic equations are also
used when gravity is involved, such as the path
of a ball or the shape of cables in a suspension
bridge.
4
Balls, Arrows, Missiles and Stones
  • If you throw a ball (or shoot an arrow, fire a
    missile or throw a stone) it will go up into the
    air, slowing down as it goes, then come down
    again ...
  • ... and a Quadratic Equation tells you where it
    will be!

5
Example Throwing a Ball
  • A ball is thrown straight up, from 3 m above the
    ground, with a velocity of 14 m/s. When does it
    hit the ground?
  • Ignoring air resistance, we can work out its
    height by adding up these three things
  • The height starts at 3 m   3
  • It travels upwards at 14 meters per second (14
    m/s)   14t
  • Gravity pulls it down, changing its speed by
    about 5 m/s per second (5 m/s2)   -5t2  

6
Add them up and the height h at any time t is
  • h-5t2 14t 3
  • And the ball will hit the ground when the height
    is zero
  • -5t2 14t 3 0
  • It will be easier to solve if we factor out the
    negative 1
  • -1(5t2 - 14t - 3 ) 0
  • Divide both sides by -1 and we have

7
5t2 - 14t - 3 0
  • 5t2 - 14t 3 (5t 1)(t 3)
  • So our two solutions are
  • t -0.2 and t 3
  • The "t -0.2" is a negative time, impossible in
    our case.
  • The "t 3" is the answer we want
  • The ball hits the ground after 3 seconds!

8
What does it look like?
  • Here is the graph of the Parabola h -5t2 14t
    3
  • It shows you the height of the ball vs time
  • Some interesting points
  • (0,3) When t0 (at the start) the ball is at 3 m
  • (-0.2,0) Says that -0.2 seconds BEFORE we threw
    the ball it was at ground level ... this never
    happened, so our common sense says to ignore it!
  • (3,0) Says that at 3 seconds the ball is at
    ground level.
  • Note also that the ball reaches nearly 13 meters
    high.

9
What about that maximum?
  • You can find exactly where the top point is!
  • Find where (along the horizontal axis) the top
    occurs using -b/2a
  • t -b/2a -(-14)/(2 5) 14/10 1.4 seconds
  • Then find the height using that value (1.4)
  • h -5t2 14t 3 -5(1.4)2 14 1.4 3
    12.8 meters
  • So the ball reaches the highest point of 12.8
    meters after 1.4 seconds.

10
Lets take a look at some more examples
  • Profit
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