Question

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?

Answer 1

Answer: 8 seconds

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Step-by-step explanation:

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

Replace h with 0 and solve for t.

`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\\\\`

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.

Answer 2

2 seconds

Step-by-step explanation:

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.