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?
Question content area bottom
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.)
Question content area bottom
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).**
\[
S(t) = 1212 + 6464t - 16t^2
\]
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).**