The flight of a water balloon is modeled by the function h=−2t2+8t+2 where h is the height of the balloon above the ground after t seconds.

Use the vertex formula to determine how many seconds it will take the water balloon to reach the maximum height. Then find the maximum height of the water balloon.

(2 points)
The water balloon will reach its maximum height at
seconds.

The maximum height of the water balloon will be
feet.

1 answer

To find the time \( t \) when the water balloon reaches its maximum height, we can use the vertex formula for a quadratic function in the form \( h = at^2 + bt + c \). The time at which the maximum height occurs is given by:

\[ t = -\frac{b}{2a} \]

In your equation \( h = -2t^2 + 8t + 2 \):

  • \( a = -2 \)
  • \( b = 8 \)

Plugging these values into the vertex formula:

\[ t = -\frac{8}{2 \cdot -2} = -\frac{8}{-4} = 2 \]

So, the water balloon will reach its maximum height at 2 seconds.

Next, we'll find the maximum height by substituting \( t = 2 \) back into the function \( h(t) \):

\[ h(2) = -2(2^2) + 8(2) + 2 \]

Calculating this:

\[ h(2) = -2(4) + 16 + 2 \] \[ = -8 + 16 + 2 \] \[ = 10 \]

Thus, the maximum height of the water balloon will be 10 feet.

Final answers:

  • The water balloon will reach its maximum height at 2 seconds.
  • The maximum height of the water balloon will be 10 feet.