Question

If a ball is thrown upward at 9696 feet per second from a height of 44 ​feet, the height of the ball can be modeled by Sequals=44plus+9696tminus−16t squaredt2 ​feet, where t is the number of seconds after the ball is thrown. How long after the ball is thrown is the height 112112 ​feet?
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Part 1
It takes enter your response here seconds for the ball to reach the height 112112 feet.
​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)

Answer 1

To find out how long after the ball is thrown it reaches a height of 112112 feet, we start with the height equation:

\[
S(t) = 44 + 9696t - 16t^2
\]

We set \( S(t) \) equal to 112112:

\[
112112 = 44 + 9696t - 16t^2
\]

Now, rearranging the equation gives us:

\[
0 = -16t^2 + 9696t + 44 - 112112
\]

Simplifying further:

\[
0 = -16t^2 + 9696t - 112068
\]

To make calculations easier, we can multiply the equation by -1:

\[
0 = 16t^2 - 9696t + 112068
\]

Now we can solve this quadratic equation \( 16t^2 - 9696t + 112068 = 0 \) using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = 16 \)
- \( b = -9696 \)
- \( c = 112068 \)

Calculating the discriminant (\( b^2 - 4ac \)):

\[
b^2 = (-9696)^2 = 93931376
\]
\[
4ac = 4 \times 16 \times 112068 = 7161088
\]
\[
b^2 - 4ac = 93931376 - 7161088 = 86770288
\]

Now we compute:

\[
t = \frac{9696 \pm \sqrt{86770288}}{32}
\]

Calculating \( \sqrt{86770288} \):

\[
\sqrt{86770288} \approx 2944.42
\]

Now substituting back into the formula:

\[
t = \frac{9696 \pm 2944.42}{32}
\]

Calculating the two possible values for \( t \):

1. Using the positive root:
\[
t_1 = \frac{9696 + 2944.42}{32} \approx \frac{12640.42}{32} \approx 394.38
\]

2. Using the negative root:
\[
t_2 = \frac{9696 - 2944.42}{32} \approx \frac{6751.58}{32} \approx 210.36
\]

Thus, the time it takes for the ball to reach the height of 112112 feet is approximately:

\[
t \approx 394.38, 210.36
\]

For completeness, depending on the context, it is usually only the positive time that is considered meaningful in these problems.

Therefore, the answer for \( t \) is approximately:

**394.38 seconds and 210.36 seconds**.

So you would need to state:

**It takes approximately 394.38 seconds (or 210.36 seconds) for the ball to reach the height of 112112 feet.**