The model for the height of the balloon captured from the ground with a balloon launcher can be represented by the function… how long will it take for the balloon to hit the ground after it’s launched?

To find the time it takes for the balloon to reach the ground, you need to set the equation equal to 0. This is because the ground technically has a height of 0.

h(t) = -16² + 128t <----- Original expression

0 = -16² + 128t <----- Plug 0 in for h(t)

16² = 128t <----- Add 16² to both sides

256 = 128t <----- Solve 16²

2 = t <----- Divide both sides by 128

Therefore, if t = 2, it will take the balloon 2 seconds to reach the ground.

jimthompson5910 jimthompson5910 · Answer 2 · 2022-07-13T04:20:18+02:00

Answer: 8 seconds

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Explanation:

The balloon is on the ground when the height is zero.

Replace h with 0 and solve for t.

[tex]h(t) = -16t^2 + 128t\\\\0 = -16t^2 + 128t\\\\-16t^2 + 128t = 0\\\\-16t(t - 8) = 0\\\\-16t = 0 \ \text{ or } \ t-8 = 0\\\\t = 0/(-16) \ \text{ or } \ t = 0+8\\\\t = 0 \ \text{ or } \ t = 8\\\\[/tex]

Ignore t = 0 because this is when the balloon is initially on the ground. In other words, the balloon starts on the ground, so it makes sense that t = 0 leads to h = 0.

The other solution t = 8 is what we're after. The balloon touches the ground again at the 8 second mark. This is the length of time the balloon is in the air.

A quick way to determine and confirm the answer is to use graphing software. Check out the diagram below. We have a parabola with x intercepts of 0 and 8.