Question

If a ball is thrown upward at 6464 feet per second from a height of 1212 ​feet, the height of the ball can be modeled by Sequals=1212plus+6464tminus−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 7272 ​feet?
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Part 1
It takes enter your response here seconds for the ball to reach the height 7272 feet.
​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)

Answers

GPT-4o mini
To find how long it takes for the ball to reach a height of 7272 feet, we can use the given equation for the height of the ball:

\[
S(t) = 1212 + 6464t - 16t^2
\]

We want to find \( t \) such that \( S(t) = 7272 \). Thus, we set up the equation:

\[
7272 = 1212 + 6464t - 16t^2
\]

Subtract 1212 from both sides:

\[
7272 - 1212 = 6464t - 16t^2
\]

This simplifies to:

\[
6060 = 6464t - 16t^2
\]

Rearranging gives us:

\[
16t^2 - 6464t + 6060 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula:

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

Here, \( a = 16 \), \( b = -6464 \), and \( c = 6060 \). Now we calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = (-6464)^2 - 4 \cdot 16 \cdot 6060
\]

Calculating \( (-6464)^2 \):

\[
(-6464)^2 = 41717456
\]

Now calculating \( 4 \cdot 16 \cdot 6060 \):

\[
4 \cdot 16 = 64
\]
\[
64 \cdot 6060 = 387840
\]

Now, substituting into the discriminant:

\[
b^2 - 4ac = 41717456 - 387840 = 41329616
\]

Now we can compute \( \sqrt{41329616} \):

\[
\sqrt{41329616} = 6428
\]

Now substituting this back into the quadratic formula:

\[
t = \frac{-(-6464) \pm 6428}{2 \cdot 16}
\]

This simplifies to:

\[
t = \frac{6464 \pm 6428}{32}
\]

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

1. \( t = \frac{6464 + 6428}{32} = \frac{12892}{32} = 403.5 \)
2. \( t = \frac{6464 - 6428}{32} = \frac{36}{32} = 1.125 \)

Thus, the ball reaches the height of 7272 feet at:

- \( t = 1.125 \) seconds, and
- \( t = 403.5 \) seconds.

Therefore, it takes **1.125 seconds** for the ball to first reach the height of 7272 feet.

So the final answer is:

**It takes 1.125 seconds for the ball to reach the height 7272 feet (and also 403.5 seconds later on).**