Question 162782
initial velocity = 200 feet / second.
h = height of the rocket in feet.
t = number of seconds after launch.
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h = -16t^2 + 200t
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the question is: in how many seconds will the rocket be 300 feet above the ground.
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since h = -16t^2 + 200t, then if we want h to be 300 feet above the ground, the equation becomes
300 = -16t^2 + 200t
subtracting 300 from both sides of the equation and it becomes
-16t^2 + 200t - 300 = 0
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can't really see which factors would work by looking at the equation, so i will use the quadratic formula.
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for the quadratic formula
a = -16
b = 200
c = -300
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b = 200
b^2 = 40000
4ac = -16 * 4 * -300 = 19200
2a = -32
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{{{(-b + sqrt (b^2-4*a*c))/(2a)}}} becomes
{{{(-200 + sqrt (40000-19200))/(-32)}}}
which becomes
t = 1.733060906 seconds.
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{{{(-b - sqrt (b^2-4*a*c))/(2a)}}} becomes
{{{(-200 - sqrt (40000-19200))/(-32)}}}
which becomes
t = 10.75693909
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t can be either 1.73.... or 10.75693.....
substituting in the original equation shows the following
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using t = 1.733060906:
300 = -16 * t^2 + 200t becomes
300 = -16*(1.733060906)^2 + 200 * (1.733060906)
this becomes 300 = 300
looks like 17.3... is a good solution.
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using t = 10.75693909
300 = -16 * (10.75693909)^2 + 200 * (10.75693909)
this becomes 300 = 300
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it looks like t could be either
1.733060906
or
10.75693909
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since the initial velocity is 200 feet / second, we can calculate how far the rocket traveled in either of those seconds assuming that velocity is stable until the rocket has at least reached 300 feet.
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taking 1.733..., the rocket will have traveled 200 feet per second * 1.733... seconds = 346.6 feet
in order for it to reach 300 feet it would have to have been aimed almost straight up.
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taking 10.756..., the rocket will have traveled 200 feet per second * 10.756... seconds = 2151 feet.
this allows a much more shallow trajectory for it to reach 300 feet.
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both answers are plausible depending on the trajectory of the rocket and the velocity being at least as fast as when it was initially launched.
answers are:
t can be either...
1.733060906
or
10.75693909
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