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sleet_krkn [62]
2 years ago
11

A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by th

e given equation. Using this equation, find the time that the rocket will hit the ground, to the nearest 100th of a second.
f(x) = -16x^2+272x+110
Mathematics
1 answer:
Bond [772]2 years ago
4 0

Answer:

The time that the rocket will hit the ground will be:

  • x = 17.40 seconds

Step-by-step explanation:

Given the expression

f\left(x\right)\:=-16x^2+272x+110

Here:

  • x represents the time in seconds
  • y represents the height of the rocket

We know that when the rocket will hit the ground, the height will get y = 0

i.e.

0\:=-16x^2+272x+110

solving for x to determine the time that the rocket takes when it will hit the ground.

so

-16x^2+272x+110=0

subtract 110 from both sides

-16x^2+272x+110-110=0-110

-16x^2+272x=-110

Divide both sides by -16

\frac{-16x^2+272x}{-16}=\frac{-110}{-16}

x^2-17x=\frac{55}{8}

\mathrm{Add\:}\left(-\frac{17}{2}\right)^2\mathrm{\:to\:both\:sides}

x^2-17x+\left(-\frac{17}{2}\right)^2=\frac{55}{8}+\left(-\frac{17}{2}\right)^2

x^2-17x+\left(-\frac{17}{2}\right)^2=\frac{633}{8}

\left(x-\frac{17}{2}\right)^2=\frac{633}{8}

\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}

solving

x-\frac{17}{2}=\sqrt{\frac{633}{8}}

x=\frac{\sqrt{1266}}{4}+\frac{17}{2}

x = 17.40 seconds

also solving

x-\frac{17}{2}=-\sqrt{\frac{633}{8}}

x=-\frac{\sqrt{1266}}{4}+\frac{17}{2}

x = -0.40 seconds

As time can not be negative.

Thus, the value of x = 17.40 seconds

Therefore, the time that the rocket will hit the ground will be:

  • x = 17.40 seconds
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