The height \( H \) of a projectile launched from ground level is a function of the time \( t \) it is in the air. The height in feet for \( t \) seconds is given by the function \( H(t)=-16 t^{2}+96 t \). What is the domain of the function? your answer using interval notation.) What does the domain mean in the context of the problem? After the projectile is launched from ground level it takes

The height function is given by \[ H(t) = -16t^2 + 96t. \] **Step 1.** Find when the projectile is at ground level by setting \( H(t) = 0 \): \[ -16t^2 + 96t = 0. \] **Step 2.** Factor the equation: \[ -16t(t - 6) = 0. \] **Step 3.** Solve \( -16t = 0 \) and \( t - 6 = 0 \): - From \( -16t = 0 \), we get \( t = 0 \). - From \( t - 6 = 0 \), we get \( t = 6 \). These times correspond to when the projectile is launched (\( t = 0 \)) and when it lands (\( t = 6 \)). **Step 4.** Since time \( t \) must be nonnegative and the projectile is only in the air between launch and landing, the domain is: \[ [0, 6]. \] **Interpretation:** In the context of the problem, the domain \([0, 6]\) means that the height function \( H(t) \) is valid only for times \( t \) between \( 0 \) seconds (at launch) and \( 6 \) seconds (when the projectile returns to the ground). Any value of \( t \) outside this interval would not represent the physical flight of the projectile.