The function h(t)=-16t^2+32t+128 is a function giving the height above a lake, in feet, of a pebble thrown off the edge of a rock cliff, where t is the time in seconds after the pebble is thrown.

Question

1. What is the initial height of the pebble before it is thrown?
2. Find the vertex of the height function. Interpret the meaning of the x- and y-coordinates of the vertex in the context of the problem.
3. At what time does the pebble hit the surface of the lake? Explain how you determined your answer. Be specific on how you used the function.
4. At t=3 seconds, is the pebble increasing or decreasing in the height above the water? Explain how you arrived at this conclusion.

This is an essay exam practice and I really need help understanding this.

Answer 1

Without "t" I don't know.

#4. Let me put it to you this way: At 3 seconds the pebble dropped (t^2+t)/2*32.2=193.2ft.

Answer 2

What is the initial height of the pebble before it is thrown? This would be at time t=0 Plug this into the equation and solve.

h(t)=-16(0)²+32(0)+128 = 128 ft is the initial height.

Find the vertex of the height function. Interpret the meaning of the x- and y-coordinates of the vertex in the context of the problem. The vertex can be found by the equation x=-b/2a

x=-32/(2(-16)) = 1 Now that we have the x coordinate, plug this into the equation to get y

h(1)=-16(1)²+32(1)+128 = 144

The vertex is (1,144) This means that the pebble hits its maximum height at a time of 1 second. The max height it reached is 144 ft.

At what time does the pebble hit the surface of the lake? Explain how you determined your answer. Be specific on how you used the function.

It hits the lake when h=0, plug this into the equation and solve.

0=-16x²+32x+128

You could solve this by factoring or the quadratic equation. Those are lengthy, the easiest way is to graph it on the calculator and get the x intercept that way. It hits the ground after 4 seconds.

At t=3 seconds, is the pebble increasing or decreasing in the height above the water? Explain how you arrived at this conclusion.

At 3 seconds it is past the vertex, therefore the height is decreasing.