To represent the height of the ball over time, we can use the equation of motion under the influence of gravity. The height \( h(t) \) of the ball at time \( t \) can be expressed as:
\[ h(t) = -16t^2 + vt + h_0 \]
where:
- \( v \) is the initial velocity (64 feet per second),
- \( h_0 \) is the initial height (0 feet, since it is kicked from the ground),
- and the term \( -16t^2 \) accounts for the acceleration due to gravity (32 feet/sec², hence the negative coefficient).
Given that Hakeem kicks the ball with an initial velocity of 64 feet per second, we have:
\[ h(t) = -16t^2 + 64t + 0 \] \[ h(t) = -16t^2 + 64t \]
We want to find the conditions when the height \( h(t) \) is less than 24 feet:
\[ -16t^2 + 64t < 24 \]
To form the quadratic inequality, we'll first rearrange the terms:
\[ -16t^2 + 64t - 24 < 0 \]
Now, we can multiply the entire inequality by -1 (which reverses the inequality sign):
\[ 16t^2 - 64t + 24 > 0 \]
Thus, the quadratic inequality that represents when the ball is less than 24 feet above the ground is:
\[ 16t^2 - 64t + 24 > 0 \]
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