when a baseball is hit by a batter, the height of the ball, h(t), at time t, t ≥ 0, is determined by the…

Explanation:

Step1: Set up the inequality

We need to find when ( h(t) \geq 52 ). Given ( h(t)= - 16t^{2}+64t + 4 ), so we set up the inequality: ( -16t^{2}+64t + 4\geq52 ) Subtract 52 from both sides: ( -16t^{2}+64t + 4-52\geq0 ) ( -16t^{2}+64t - 48\geq0 ) Divide both sides by - 16 (remember to reverse the inequality sign): ( t^{2}-4t + 3\leq0 )

Step2: Factor the quadratic

Factor the quadratic expression ( t^{2}-4t + 3 ). We need two numbers that multiply to 3 and add to - 4. The numbers are - 1 and - 3. So, ( t^{2}-4t + 3=(t - 1)(t - 3) ) So the inequality becomes ( (t - 1)(t - 3)\leq0 )

Step3: Find the critical points and test intervals

The critical points are ( t = 1 ) and ( t = 3 ) (where the expression equals zero). We test the intervals:

• For ( t<1 ), let's pick ( t = 0 ). Then ( (0 - 1)(0 - 3)=(-1)\times(-3) = 3>0 ), which does not satisfy ( (t - 1)(t - 3)\leq0 )
• For ( 1<t<3 ), let's pick ( t = 2 ). Then ( (2 - 1)(2 - 3)=(1)\times(-1)=- 1\leq0 ), which satisfies the inequality. But we also need to check the endpoints.
• For ( t>3 ), let's pick ( t = 4 ). Then ( (4 - 1)(4 - 3)=(3)\times(1)=3>0 ), which does not satisfy the inequality.

Now check the endpoints: When ( t = 1 ), ( (1 - 1)(1 - 3)=0\times(-2)=0\leq0 ) When ( t = 3 ), ( (3 - 1)(3 - 3)=2\times0 = 0\leq0 )

So the solution to ( (t - 1)(t - 3)\leq0 ) is ( 1\leq t\leq3 )

Answer:

( 1\leq t\leq3 ) (corresponding to the option D. ( 1\leq t\leq3 ))