Question 820472
(a)
The equation is:
{{{ y(x) = -16x^2 + 320x }}}
This is a parabola with a maximum that
begins at ( x,y ) = ( 0,0 )
The squared term is entirely the
effect of gravity pulling the rocket down
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(b)
The x-coordinate of the vertex is at
{{{ -b/(2a) }}}
{{{ a = -16 }}}
{{{ b = 320 }}}
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{{{ -b/(2a) = -320 / ( 2*(-16) ) }}}
{{{ -b/(2a) = -320/(-32) }}}
{{{ -b/(2a) = 10 }}}
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To find the y-coordinate of the vertex, plug
this value of {{{ x }}} back into the equation
{{{ y(x) = -16x^2 + 320x }}}
{{{ y(10) = -16*(10)^2 + 320*(10) }}}
{{{ y(10) = -1600 + 3200 }}}
{{{ y(10) = 1600 }}}
The maximum height is 1600 ft
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(c)
The rocket took 10 sec to reach the maximum height
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(d)
You have to find out when the height, {{{ y(x) }}},
is zero again, so you have to find both roots
{{{ y(x) = -16x^2 + 320x }}}
{{{  -16x^2 + 320x = 0 }}}
{{{ x*( -16x + 320 ) = 0 }}}
{{{ -16x + 320 = 0 }}}
{{{ 16x = 320 }}}
{{{ x = 20 }}}
The rocket took 20 sec to come back to earth
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(e)
The velocity when it hit the ground is the same as it
was at takeoff, 320 ft/sec
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Here's a plot of the equation:
{{{ graph( 400, 400, -4, 25, -200, 2000, -16x^2 + 320x ) }}}