A ball is thrown in the air from the top of a 50-foot high building .h(t) is a function that gives the height of the ball from the ground , in feet , in terms of t , the time in seconds. You may assume that t=0 corresponds to the time the ball is thrown .
Which of the following equations for h is consistent with the given information ?

Options

A. `h=50+10t+16t^2`
B. `h=50+10t-16t^2`
C. `h=-50 + 10t-16t^2`
D. `h=-50+10t+16t^2`

Answer

To solve the problem, we need to find a function \( h(t) \) that describes the height of a ball thrown from a 50-foot high building in terms of time \( t \). The function must satisfy the following conditions:

1. The initial height of the ball at \( t = 0 \) must be 50 feet.
2. The height of the ball must decrease over time as it is thrown upwards and then falls back down.

### Step-by-Step Solution:

1. **Understanding the Function Form**:
The height function \( h(t) \) is typically a quadratic function of the form:
\[ h(t) = -at^2 + bt + c \]
where \( a, b, c \) are constants. The negative sign in front of \( at^2 \) indicates that the parabola opens downwards, which is necessary since the ball reaches a maximum height and then falls back down.

2. **Setting the Initial Height**:
Since the ball is thrown from a height of 50 feet, we set the initial condition:
\[ h(0) = 50 \]
Substituting \( t = 0 \) into the function gives:
\[ h(0) = -a(0)^2 + b(0) + c = c \]
Therefore, \( c = 50 \).

3. **Choosing the Correct Equation**:
We need to analyze the options provided (let's assume they are labeled A, B, C, and D) to find which one has:
- A negative coefficient for \( t^2 \) (to ensure the parabola opens downwards).
- An initial height of 50 feet when \( t = 0 \).

4. **Evaluating the Options**:
- For each option, substitute \( t = 0 \) and check if \( h(0) = 50 \).
- Check the coefficient of \( t^2 \) to ensure it is negative.

5. **Finding the Correct Option**:
After evaluating the options:
- If option B is the only one that satisfies both conditions (negative coefficient for \( t^2 \) and \( h(0) = 50 \)), then option B is the correct answer.

### Conclusion:
The function that describes the height of the ball thrown from the top of a 50-foot building is consistent with option B, which meets all the criteria.

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