PSSA - Pennsylvania System of School Assessment Edward Bujak When a cannonball is shot into the air, its height, in feet, during the flight is given by the function hHtL = 160 + 480 t - 16 t2 where hHtL is the height in seconds. What is the maximum height of the cannonball? A B C D
15 feet 160 feet 3,744 feet 3,760 feet
Define a function: h@t_D := 160 + 480 t - 16 t2 ; Factor @h@tDD - 16 I- 10 - 30 t + t2 M
Factoring does not tell us much.
When the cannon was fired, what was the height? What's the height at time, t=0. h@0D
160
Well, when does the cannonball hit that same height, h=160 feet, again? zeros = Solve @h@tD 160, tD 88t ® 0<, 8t ® 30<<
The cannonball hits the initial height (160 feet) at time t=0 seconds and t=30 seconds. It is logical that the cannonball reaches its maximum height at half the time t=15 seconds, since a parabola is symmetric. h@15D
3760
So the maximum height of the cannonball is 3,760 feet, reached at time 15 seconds (after firing). Graphically:
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PSSA.20090224.D.Parabola,Quadratic,Vertex,Maximum.nb
Graphically: parabolaCannonballShotPlot = Plot@8h@tD, h@0D, h@15D< , 8t, - 1, 32<, AxesLabel ® 8"t @secD", "hHtL @feetD"<, PlotLabel ® "Cannonball - Height vs Time", PlotStyle ® 8Blue, Dashed, Dashed
Cannonball - Height vs Time
hHtL @feetD
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t @secD
The blue parabolic curve is the trajectory of the cannonball, height vs time. The dashed horizontal lines represent the minimum height and the maximum height of the cannonball. pointsOnPlot = ListPlot@880, h@0D<, 815, h@15D<, 830, h@30D<<, PlotStyle ®
[email protected], Pink
Cannonball - Height vs Time
hHtL @feetD
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t @secD
There are other Mathematica functions that can be utilized. FindMaximum@h@tD, 8t, 5
If we do not have these tools, or do not know about them then we can resort to traditional Calculus means: that maximum or minimum occur when the derivative is set equal to 0.
PSSA.20090224.D.Parabola,Quadratic,Vertex,Maximum.nb
If we do not have these tools, or do not know about them then we can resort to traditional Calculus means: that maximum or minimum occur when the derivative is set equal to 0. h '@tD
480 - 32 t
Solve@h '@tD 0D 88t ® 15<<
So the maximum is at time, t=15 seconds, and height, h(t=15 seconds) h@15D
3760
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