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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 the given equation. Using this equation, find the time that the rocket will hit the ground, to the nearest 10oth of second. y=-16x^(2)+89 x+50

A rocket is launched from a tower. The height of the rocket, y y in feet, is related to the time after launch, x \mathrm{x} in seconds, by the given equation. Using this equation, find the time that the rocket will hit the ground, to the nearest 1010oth of second.\newliney=−16x2+89x+50 y=-16 x^{2}+89 x+50

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Full solution

Q. A rocket is launched from a tower. The height of the rocket, y y in feet, is related to the time after launch, x \mathrm{x} in seconds, by the given equation. Using this equation, find the time that the rocket will hit the ground, to the nearest 1010oth of second.\newliney=−16x2+89x+50 y=-16 x^{2}+89 x+50
  1. Find xx when y=0y=0: First, we need to find when y=0y=0, because that's when the rocket hits the ground.\newlineSo we set the equation to 00 and solve for xx: 0=−16x2+89x+500 = -16x^2 + 89x + 50.
  2. Use quadratic formula: Now we need to use the quadratic formula to solve for xx, which is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Here, a=−16a = -16, b=89b = 89, and c=50c = 50.
  3. Calculate discriminant: Let's calculate the discriminant first: b2−4ac=892−4(−16)(50)b^2 - 4ac = 89^2 - 4(-16)(50). That's 7921+32007921 + 3200, which equals 1112111121.
  4. Take square root: Now we take the square root of the discriminant: 11121=105.455\sqrt{11121} = 105.455.
  5. Plug values into formula: We can now plug the values into the quadratic formula: x=−89±105.4552⋅−16x = \frac{{-89 \pm 105.455}}{{2 \cdot -16}}.
  6. Calculate first solution: We have two possible solutions for xx: x=−89+105.455−32x = \frac{{-89 + 105.455}}{{-32}} and x=−89−105.455−32x = \frac{{-89 - 105.455}}{{-32}}.
  7. Calculate second solution: Calculating the first solution: x=(16.455)/−32=−0.514x = (16.455) / -32 = -0.514. This doesn't make sense because time can't be negative.
  8. Round to nearest hundredth: Calculating the second solution: x=(−194.455)/−32=6.076x = (-194.455) / -32 = 6.076. This is the time when the rocket will hit the ground.
  9. Round to nearest hundredth: Calculating the second solution: x=(−194.455)/−32=6.076x = (-194.455) / -32 = 6.076. This is the time when the rocket will hit the ground.Finally, we round the time to the nearest hundredth of a second: 6.086.08 seconds.

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