Quadratic-Story Problem?

Question

An object is propelled vertically upward from the top of a 112-foot building.
The quadratic function s(t) = -16t² + 176t + 112 models the ball's height above the ground, s(t), in feet, t seconds after it was thrown.
How many seconds does it take until the object finally hits the ground? Round to the nearest tenth of a second if necessary.

Answer 1

On the ground, s(t) = 0;
-16tt + 176t + 112 = 0.
Divide by -16:
tt - 11t - 7 = 0.
Use formula;
t = 11/2 +/- [sqrt(121 + 28)]/2.
Positive answer [neasrest tenth of sec]
11.6s

Answer 2

It will take about 11.6 seconds for the object to hit the ground.

What we want to do is set the given equation equal to 0, because after t seconds the object will be at 0 height. To simplify calculations, divide through by 16 first, because both the coefficients of the variable terms and the constant are divisible by 16 in this problem. Doing this won't change the answer, because 0 divided by 16 is still equal to 0.

s (t) = 0 = (-16t² + 176t + 112 ) / 16 = -t² + 11t + 7.

We can see by looking at the equation on the right that it isn't factorable, because no sum or difference of the factors of 7 will add up to 11.

Since s(t) = 0 = -t² + 11t + 7 is a quadratic function, we can use the quadratic formula to determine t. The quadratic formula says this:

Given an equation in the form ax² + bx + c,
x = -b ± √(b² - 4ac) / 2a.

Now, simply plug your values for a, b and c into the quadratic formula and crank out the answer. In this problem, a = -1, b = 11, and c = 7. We can use the variable name t instead of x for purposes of calculating the solution.

t = {-11 ± √[(11)² - (4)(-1)(7)]} / [2 (-1)]
t = [-11 ± √(121 + 28)] / -2
t = (-11 ± √149) / -2
t = (-11 ± 12.207) / -2
t = (-11 + 12.207) / -2 or t = (-11 - 12.207) / -2.

We can see by visually inspecting the possible solutions that we can reject the first one, because t will be a positive number divided by a negative number, resulting in a negative value for t. Since we have yet to discover a way to reverse time, that solution is a no go. That leaves us with this:

t = (-11 - 12.207) / -2
t = -23.207 / -2
t = 11.6035 sec, which we can round to t = 11.6 sec.

Answer 3

Plugging into the quadratic formula gives me 11.6 seconds.