Answer:

Let's solve both problems step by step:

3. The equation for the height of the pen as a function of time is given as h(t) = -16t^2 + 250, where t is the time in seconds, and h is the height in feet.

To find the height of the pen 3 seconds after it was dropped, substitute t = 3 into the equation:

h(3) = -16(3^2) + 250

h(3) = -16(9) + 250

h(3) = -144 + 250

h(3) = 106 feet

So, the height of the pen 3 seconds after it was dropped is 106 feet.

4. The equation for the height of the rock above the water as a function of time is given as f(t) = -16t^2 + 32t + 80, where t is the time in seconds, and f(t) is the height in feet.

To find the maximum height of the rock above the water, we need to determine the vertex of this quadratic equation. The vertex can be found using the formula:

t_vertex = -b / (2a)

In this equation, a = -16 and b = 32. Plug these values into the formula:

t_vertex = -32 / (2 * (-16))

t_vertex = -32 / (-32)

t_vertex = 1

Now that we have the time when the rock reaches its maximum height, we can find the maximum height itself by substituting t = 1 into the equation:

f(1) = -16(1^2) + 32(1) + 80

f(1) = -16 + 32 + 80

f(1) = 96 feet

So, the maximum height of the rock above the water is 96 feet.