When a base ball by a batter, the height of the ball, h(t), at time t, is determined by the equation `h(t)=-16t^(2)+64t+4`, where `tge0`. For which interval of time, in seconds, is the height of the ball at least 52 feet above the playing field?

Options

A. `0.5letle2.5`
B. `1.0letle3.0`
C. `1.5letle3.5`
D. `2.0letle4.0`

Answer

To solve the problem, we need to determine the interval of time \( t \) during which the height of the baseball \( h(t) \) is at least 52 feet. The height of the baseball is given by the equation:

\[ h(t) = -16t^2 + 64t + 4 \]

We need to find when \( h(t) \geq 52 \).

### Step 1: Set up the inequality
We start by setting up the inequality:

\[ -16t^2 + 64t + 4 \geq 52 \]

### Step 2: Rearrange the inequality
Next, we rearrange the inequality by subtracting 52 from both sides:

\[ -16t^2 + 64t + 4 - 52 \geq 0 \]

This simplifies to:

\[ -16t^2 + 64t - 48 \geq 0 \]

### Step 3: Factor out common terms
We can factor out -16 from the left side:

\[ -16(t^2 - 4t + 3) \geq 0 \]

Dividing the entire inequality by -16 (and remembering to reverse the inequality sign):

\[ t^2 - 4t + 3 \leq 0 \]

### Step 4: Factor the quadratic expression
Now, we factor the quadratic expression:

\[ (t - 1)(t - 3) \leq 0 \]

### Step 5: Determine the critical points
The critical points from the factors are \( t = 1 \) and \( t = 3 \).

### Step 6: Test intervals
We will test the intervals determined by the critical points:

1. **Interval 1**: \( t < 1 \) (e.g., \( t = 0 \)):
\[ (0 - 1)(0 - 3) = ( -1)( -3) = 3 > 0 \]

2. **Interval 2**: \( 1 < t < 3 \) (e.g., \( t = 2 \)):
\[ (2 - 1)(2 - 3) = (1)(-1) = -1 < 0 \]

3. **Interval 3**: \( t > 3 \) (e.g., \( t = 4 \)):
\[ (4 - 1)(4 - 3) = (3)(1) = 3 > 0 \]

### Step 7: Conclusion of intervals
The inequality \( (t - 1)(t - 3) \leq 0 \) holds true in the interval:

\[ 1 \leq t \leq 3 \]

Thus, the height of the ball is at least 52 feet during the interval \( [1, 3] \).

### Final Answer
The interval of time in seconds during which the height of the ball is at least 52 feet is:

\[ \text{Option B: } 1 \leq t \leq 3 \]