The height of a boulder launched from a Roman catap can be described as a function of time according to the following quadratic equation: `h(t)=-16 t ^(2) +224t+240.`
What is the maximum height that the boulder attains ?

Options

A. 240
B. 784
C. 1024
D. 1696

Explanation

To find the maximum height that the boulder attains, we will follow these steps:

### Step 1: Identify the quadratic function
The height of the boulder as a function of time is given by:
\[ h(t) = -16t^2 + 224t + 240 \]

### Step 2: Find the derivative of the function
To find the maximum height, we need to compute the derivative of \( h(t) \) with respect to \( t \):
\[ h'(t) = \frac{d}{dt}(-16t^2 + 224t + 240) \]
Using the power rule, we get:
\[ h'(t) = -32t + 224 \]

### Step 3: Set the derivative equal to zero
To find the critical points where the maximum height occurs, we set the derivative equal to zero:
\[ -32t + 224 = 0 \]

### Step 4: Solve for \( t \)
Rearranging the equation gives:
\[ 32t = 224 \]
\[ t = \frac{224}{32} \]
\[ t = 7 \]

### Step 5: Substitute \( t \) back into the original function
Now that we have \( t = 7 \), we substitute this value back into the original height function to find the maximum height:
\[ h(7) = -16(7^2) + 224(7) + 240 \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now substituting:
\[ h(7) = -16(49) + 224(7) + 240 \]
\[ h(7) = -784 + 1568 + 240 \]

### Step 6: Simplify the expression
Now we simplify:
\[ h(7) = -784 + 1568 + 240 \]
Calculating \( -784 + 1568 \):
\[ -784 + 1568 = 784 \]
Now adding \( 240 \):
\[ 784 + 240 = 1024 \]

### Conclusion
The maximum height that the boulder attains is:
\[ \boxed{1024} \]