Athena W.

asked • 10/04/16

Problem solving- Complex Numbers and Roots

At a carnival, A new attraction allows contestants to jump off a springboard onto a platform to be launched vertically into the air.  The object is to ring a bell located 20 feet overhead.  The distance d(t)=16t^2-bt+20, where t is the time in seconds after leaving the platform, and b is the takeoff velocity from the platform. 
 
1.  Kate watches some of the contestants. She theorizes that if the platform lanches a contestant with a takeoff velocity of at least 32 ft./s the contestant can ring the bell.
a. Find the zeros for the function using 32 feet per second as the takeoff velocity. 
B. Is Kate's theory valid?

Arturo O.

Are you sure the equation for d(t) is correct?  While rising up, the acceleration is down, which would make the equation come out to be
 
d = -16t2 + bt + 20
 
Note also the b in front of t is positive if launched upward with up direction defined as positive.
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10/04/16

Arturo O.

In addition, the constant term would have to be the height of the platform from which the person jumps.
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10/04/16

Athena W.

on the sheet with the problem the 16 is a positive. It was one of the things that made this problem confusing to me
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10/04/16

Arturo O.

I would suspect the +16 is a typo, unless it is intentional and part of the question.  The -16 comes from
 
d(t) = (1/2)at2 + v0t + d0
 
a = -g = -32 ft/s2
 
v0 = initial vertical velocity imparted by the platform, v0 positive up
 
d0 = height of platform (assume it is zero?)
 
Then
 
d(t) = -16t2 + v0t
 
Assume v0 = 32 ft/s, and you want to reach a height of 20 ft.  Is there a time t when this is realized?
 
20 = -16t2 + 32t
 
-16t2 + 32t - 20 = 0
 
Multiply both sides by -1 and get
 
16t2 - 32t + 20 = 0
 
Divide by 4 to simplify a little.
 
4t2 - 8t + 5 = 0
 
t = [8 ± √(64 - 4*4*5)] / 8
 
Note you get a complex number, so a launch speed of 32 ft/s will not get you up to 20 ft.
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10/04/16

Athena W.

Okay. thank you
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10/04/16

1 Expert Answer

By:

Arturo O. answered • 10/04/16

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