- Identify the height function \(h(t) = -16t^2 + 80t\) and set it to zero to find the time when the ball reaches the ground.
- Factor the quadratic equation \(-16t^2 + 80t = 0\) to get \(-16t(t-5) = 0\).
- Solve for \(t\) to find the roots \(t = 0\) and \(t = 5\).
- Discard \(t=0\) as the initial time and conclude the ball reaches the ground at \(\boxed{5}\) seconds.
Explanation
- Analysis of the Problem
The height \(h\) of the ball in feet at time \(t\) in seconds is given by the function \(h(t) = -16t^2 + 80t\). We are looking for the time \(t\) when the ball reaches the ground, which corresponds to the height \(h = 0\). We are given that \(t > 0\) since the ball is thrown into the air at \(t = 0\).
- Solving the Equation
To find when the ball hits the ground, we set the height function equal to zero:
\[ -16t^2 + 80t = 0 \]
We can factor out the common term \(-16t\) from the left side of the equation:
\[ -16t(t - 5) = 0 \]
This gives us two possible solutions for \(t\):
\[ -16t = 0 \quad \text{or} \quad t - 5 = 0 \]
Solving these linear equations, we get \(t = 0\) or \(t = 5\). The solution \(t = 0\) represents the moment the ball is thrown, and \(t = 5\) represents the time when the ball returns to the ground.
- Conclusion
Since the question asks for the time after which the ball reaches the ground, we identify \(t = 5\) seconds as the correct time. Comparing this result with the given options, it corresponds to option (c). Thus, the ball reaches the ground after 5 seconds.
Examples
Understanding projectile motion is essential in sports and engineering. For example, if a professional athlete throws a ball, coaches use these physics models to analyze the trajectory, hang time, and the optimal angle for the throw. By calculating when the ball lands, athletes can better predict where to position themselves for a catch or how much force is needed for a specific distance. This math foundation helps in optimizing performance in various fields like athletics, aviation, and game design.