To solve this problem, we need to find the time when the pebble hits the ground, which is when the height of the pebble (h) becomes 0.
Given information:
- The height of the waterfall is 1100 feet.
- The initial velocity of the pebble is 16 feet per second.
- The height of the pebble (h) in feet after t seconds is given by the equation:
h = -16t^2 + 16t + 1100
Step 1: Set the height of the pebble (h) equal to 0 to find the time when it hits the ground.
0 = -16t^2 + 16t + 1100
Step 2: Solve the quadratic equation for t.
-16t^2 + 16t + 1100 = 0
t^2 - t - 68.75 = 0
Using the quadratic formula:
t = (-(-1) ± √((-1)^2 - 4 × (-16) × 1100)) / (2 × (-16))
t = (1 ± √(1 + 70400)) / (-32)
t = (1 ± √70401) / (-32)
There are two solutions for t, but we are interested in the positive value, which represents the time when the pebble hits the ground.
t = (1 + √70401) / (-32)
t = (1 + 265.33) / (-32)
t = 266.33 / (-32)
t = -8.32 seconds
Therefore, the pebble will hit the ground 8.32 seconds after it is thrown.
