A stone projected vertically upward with initial velocity of 112 feet per second moves according to the equation
`s=112t-16t^2`
where s is the distance , in feet , from the ground , and t is time , in seconds. What is the maximum height reached by the stone ?

Options

A. 672 feet
B. 196 feet
C. 112 feet
D. 96 feet

Answer

To find the maximum height reached by the stone projected vertically upward, we can follow these steps:

### Step 1: Understand the given equation
The distance \( s \) (in feet) from the ground as a function of time \( t \) (in seconds) is given by the equation:
\[ s = 112t - 16t^2 \]

### Step 2: Find the velocity equation
To find the maximum height, we first need to determine when the stone stops rising, which occurs when the velocity is zero. The velocity \( v \) is the derivative of the distance \( s \) with respect to time \( t \):
\[ v = \frac{ds}{dt} = 112 - 32t \]

### Step 3: Set the velocity to zero
To find the time \( t \) at which the stone reaches its maximum height, we set the velocity equation to zero:
\[ 112 - 32t = 0 \]

### Step 4: Solve for \( t \)
Rearranging the equation gives:
\[ 32t = 112 \]
\[ t = \frac{112}{32} = 3.5 \text{ seconds} \]

### Step 5: Substitute \( t \) back into the distance equation
Now that we have the time at which the maximum height occurs, we substitute \( t = 3.5 \) back into the distance equation to find the maximum height \( s \):
\[ s = 112(3.5) - 16(3.5)^2 \]

### Step 6: Calculate the maximum height
Calculating each term:
1. \( 112 \times 3.5 = 392 \)
2. \( 16 \times (3.5)^2 = 16 \times 12.25 = 196 \)

Now, substituting these values back into the equation:
\[ s = 392 - 196 = 196 \text{ feet} \]

### Conclusion
The maximum height reached by the stone is:
\[ \boxed{196 \text{ feet}} \]