Question

a soccer ball kicked from the ground with an initial velocity of 32 ft/s is given by the function $h(t) = -16t^2 + 32t$. when is the soccer ball moving through the air?
○ {t | -2 < t < 2}
○ {t | -4 < t < 4}
○ {t | 0 < t < 2}
○ {t | 0 < t < 4}

Explanation:

Step1: Find when the ball is on the ground

The height function is \( h(t) = -16t^2 + 32t \). The ball is on the ground when \( h(t) = 0 \). So we solve \( -16t^2 + 32t = 0 \).
Factor out \( -16t \): \( -16t(t - 2) = 0 \).
This gives solutions \( t = 0 \) (when it's kicked) and \( t = 2 \) (when it lands).

Step2: Determine time in air

Since time can't be negative in this context, the ball is moving through the air between when it's kicked (\( t = 0 \)) and when it lands (\( t = 2 \)), so \( 0 < t < 2 \).

Answer:

\(\{t\mid 0 < t < 2\}\) (the third option)