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A hovercraft takes off from a platform. Its height (in meters), x seconds after takeoff, is modeled by: h(x)=-3(x-3)^(2)+108 How many seconds after takeoff will the hovercraft land on the ground? seconds

A hovercraft takes off from a platform.\newlineIts height (in meters), \newlinexx seconds after takeoff, is modeled by:\newlineh(x)=−3(x−3)2+108h(x)=-3(x-3)^{2}+108\newlineHow many seconds after takeoff will the hovercraft land on the ground?\newlineseconds\text{seconds}

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Full solution

Q. A hovercraft takes off from a platform.\newlineIts height (in meters), \newlinexx seconds after takeoff, is modeled by:\newlineh(x)=−3(x−3)2+108h(x)=-3(x-3)^{2}+108\newlineHow many seconds after takeoff will the hovercraft land on the ground?\newlineseconds\text{seconds}
  1. Understand the Problem: Understand the problem.\newlineWe need to find the time when the hovercraft's height is 00, which means it has landed on the ground. The height is given by the function h(x)=−3(x−3)2+108h(x) = -3(x - 3)^2 + 108.
  2. Set Height Function: Set the height function equal to 00 to find when the hovercraft lands.\newline0=−3(x−3)2+1080 = -3(x - 3)^2 + 108
  3. Solve for x: Solve for x.\newlineFirst, move the constant term to the other side of the equation:\newline−3(x−3)2=−108-3(x - 3)^2 = -108
  4. Isolate Squared Term: Divide both sides by −3-3 to isolate the squared term.\newline(x−3)2=36(x - 3)^2 = 36
  5. Solve for x: Take the square root of both sides to solve for x−3x - 3.x−3=±6x - 3 = \pm 6
  6. Discard Negative Value: Solve for xx by adding 33 to both possible values of x−3x - 3.
    x=3+6x = 3 + 6 or x=3−6x = 3 - 6
    x=9x = 9 or x=−3x = -3
  7. Discard Negative Value: Solve for xx by adding 33 to both possible values of x−3x - 3.
    x=3+6x = 3 + 6 or x=3−6x = 3 - 6
    x=9x = 9 or x=−3x = -3Since time cannot be negative, we discard the negative value.
    The hovercraft will land on the ground after 99 seconds.

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