A projectile is launched from ground level with an initial velocity of v_(0) feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s=-16t^2+v_(0)t Find the time(s) that the projectile will (a)reach a height of 288 ft and (b) return to the ground when v_(0)=144 feet per second. (a) Find the time(s)that the projectile will reach a height of 288 ft when v_(0)=144 feet per second. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. D A. 3,6 seconds (Use a comma to separate answers as needed.) B. The projectile does not reach 288 feet. (b) The projectile returns to the ground after square second(s)

To solve this problem, we need to use the given equation for the height of the projectile: $s = -16t^2 + v_0t$.<br /><br />(a) To find the time(s) that the projectile will reach a height of 288 ft, we need to set $s$ equal to 288 and solve for $t$.<br /><br />$-16t^2 + 144t = 288$<br /><br />Divide both sides by -16:<br /><br />$t^2 - 9t = -18$<br /><br />Rearrange the equation:<br /><br />$t^2 - 9t + 18 = 0$<br /><br />Now, we can solve this quadratic equation using the quadratic formula:<br /><br />$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />where $a = 1$, $b = -9$, and $c = 18$.<br /><br />Plugging in the values, we get:<br /><br />$t = \frac{9 \pm \sqrt{81 - 72}}{2}$<br /><br />$t = \frac{9 \pm \sqrt{9}}{2}$<br /><br />$t = \frac{9 \pm 3}{2}$<br /><br />So, the solutions are:<br /><br />$t = \frac{9 + 3}{2} = 6$<br /><br />and<br /><br />$t = \frac{9 - 3}{2} = 3$<br /><br />Therefore, the correct answer is A. 3,6 seconds.<br /><br />(b) To find the time it takes for the projectile to return to the ground, we need to find the time when $s = 0$.<br /><br />$-16t^2 + 144t = 0$<br /><br />Factor out $t$:<br /><br />$t(-16t + 144) = 0$<br /><br />This gives us two solutions:<br /><br />$t = 0$ (which is the initial launch time)<br /><br />and<br /><br />$-16t + 144 = 0$<br /><br />Solving for $t$:<br /><br />$16t = 144$<br /><br />$t = 9$<br /><br />Therefore, the projectile returns to the ground after 9 seconds.