Find the time when the balloon hits the ground. Answer in complete sentences.

Marvin was standing on the roof of his house throwing water balloons onto people on the sidewalk below. When he let the balloon go the balloon was 45 cubits off of the ground. He learned through trial and error that in order to hit someone on the ground he had to have the balloon reach a maximum height of 55 cubits off of the ground 2 seconds after he threw it. Let h(t) be the height of the water balloon in cubits t seconds after it was thrown. Assume that height is a quadratic function of time.
a) Find formula for h(t)
b) Find the time when the balloon hits the ground. Answer in complete sentences.

Expert Answers

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For this question we can think about what we know:
At the starting time, the balloon is released at 45 cubits in height
After 2 seconds it reaches a maximum of 55 cubits off the ground

PART A) We want to write a formula h(t) to represent the height od the balloon at a given time called t. There are different equations for quadratics, one being standard form of y=ax^2 + bx + c, and one being vertex form, or y = a(x-h)^2 + k. Because we know the maximum height and time it took to get there, we have the information to plug in for the vertex (h,k) = (2, 55). Let's plug that into the vertex form formula, but we'll use t and h(t) instead of x and y:

h(t) = a(t-2)^2 + 55

Now we want to fill in whatever else we know so that we can finish our equation. We have the information of one other point in the balloon's path, which is the starting point: at a time = 0 seconds, the balloon is 45 cubits up. Let's plug this in for our time and height:

45 = a(0-2)^2 + 55

Now the only variable in our equation is a, so we can use al of this information to solve for a:

45 = a * (-2)^2 + 55

45 = a * 4 + 55

-10 = a* 4

-5/2 = a

Now let's plug this back into our vertex form equation:

h(t) = (-5/2)(t-2)^2 + 55

a) The formula for the height of the balloon at some time t is h(t) = (-5/2)(t-2)^2 + 55

 To find the time the balloon hits the ground, let's think about what that means... At a certain time t that we want to find, the balloon with be at what height? Our formula tells us the height of the balloon above the ground, so when the balloon hits the ground, it is at a height of 0. To find the time when it hits the ground, all we have to do is plug in 0 as our height and solve for the time t.

0 = (-5/2)(t-2)^2 + 55       Subtract 55

-55 = (-5/2)(t-2)^2           Divide by -5/2

22 = (t - 2)^2                   Square root both sides

sqrt(22) = t - 2                 Add 2

t = sqrt(22) + 2 = 6.69041..

b) The balloon will hit the ground after 6.69 seconds.

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