Final answer:
To determine how long the soccer ball is in the air, the quadratic equation f(x) = -8x² + 24x, representing the ball's height over time, is solved to find when the height equals zero. The ball is in the air for 3 seconds before hitting the ground.
Step-by-step explanation:
The student's question involves finding out how long a soccer ball is in the air after being kicked, assuming a quadratic function represents the ball's height over time. The function provided is f(x) = -8x² + 24x, where x is the time in seconds. The soccer ball will hit the ground when the height is zero, which occurs when f(x) = 0. To find this time, we would set the equation to zero and solve for x.
Let's solve the equation -8x² + 24x = 0. Factoring out -8x gives us -8x(x - 3) = 0. Therefore, the ttwowo solutions for x are 0 (when the ball is kicked) and 3 (when the ball hits the ground). This means the soccer ball will be in the air for 3 seconds.