benjamin threw a rock straight up from a cliff that was 96 ft above the water if the height of the rock h ,in feet, after t seconds is given by the equation h = "-16t^2" + 116t + 96

The height of the rock, \( h \), after \( t \) seconds is modeled by the equation:

\[
h(t) = -16t^2 + 116t + 96
\]

This is a quadratic equation that represents the vertical motion of the rock under the influence of gravity. Let's analyze the situation step by step.

### Key Points:
1. **Initial Height**: The initial height is the constant term, 96 ft. This is the height of the cliff from which the rock was thrown.
   
2. **Initial Velocity**: The initial velocity of the rock is given by the coefficient of \( t \), which is 116 ft/s. This indicates that Benjamin threw the rock upwards with an initial velocity of 116 feet per second.

3. **Acceleration Due to Gravity**: The term \( -16t^2 \) represents the effect of gravity, with \( -16 \) being half of the gravitational acceleration (in feet per second squared) on Earth.

### Some Important Questions to Consider:

1. **When does the rock reach its maximum height?**
   The rock will reach its maximum height when its velocity becomes zero. This happens at the vertex of the parabola.

2. **What is the maximum height of the rock?**
   You can calculate the maximum height by finding the value of \( t \) at the vertex, then substituting it back into the equation to get \( h \).

3. **When does the rock hit the water?**
   The rock will hit the water when \( h = 0 \). We can solve the quadratic equation to find the time \( t \) when the rock reaches the water.

4. **How long is the rock in the air?**
   The total time the rock is in the air can be found by solving for when \( h = 0 \).

5. **What is the velocity of the rock at any given time?**
   The velocity of the rock is given by the derivative of the height function, which can be used to calculate the speed at any time.

Would you like to go through the detailed solutions for any of these questions?

Here are some questions to expand the understanding:
1. What is the time when the rock reaches its maximum height?
2. How do you calculate the maximum height the rock reaches?
3. At what time will the rock hit the water?
4. How can you determine the initial velocity from the equation?
5. How does the acceleration due to gravity affect the motion of the rock?

**Tip**: The maximum height of any projectile thrown upwards can be easily found by using the formula \( t = -\frac{b}{2a} \) to find the vertex in the quadratic equation.

benjamin threw a rock straight up from a cliff that was 96 ft above the water if the height of the rock h, in feet, after t seconds is given by the equation h = -16t^2 + 116t + 96

Explore the quadratic equation h = -16t^2 + 116t + 96 to model the height of a rock thrown upward from a cliff 96 feet above water. Learn how to find the maximum height, time of impact, and velocity, with key algebraic and calculus concepts. Suitable for students in Grades 9-11.

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