Sure, let's solve this step-by-step.
Given the height function for the pebble as it falls:
[tex]\[ h(t) = -16t^2 + \text{initial height} \][/tex]
We are given that:
- The initial height of the cliff is 784 feet.
- The value for gravity acceleration is included in the [tex]\(-16t^2\)[/tex] term.
- The pebble will hit the ground when [tex]\( h(t) = 0 \)[/tex].
So we can rewrite the height function as:
[tex]\[ h(t) = -16t^2 + 784 \][/tex]
We need to find the time [tex]\( t \)[/tex] when the height of the pebble is 0 (i.e., when it hits the ground). Thus, we set:
[tex]\[ 0 = -16t^2 + 784 \][/tex]
To solve for [tex]\( t \)[/tex], follow these steps:
1. Move the constant term to the other side of the equation:
[tex]\[ 16t^2 = 784 \][/tex]
2. Divide both sides by 16 to isolate [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{784}{16} \][/tex]
3. Simplify the division:
[tex]\[ t^2 = 49 \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \sqrt{49} \][/tex]
5. The square root of 49 is:
[tex]\[ t = 7 \][/tex]
Therefore, it will take [tex]\( 7 \)[/tex] seconds for the pebble to hit the ground. Hence, the answer is:
[tex]\[ \boxed{7.0} \][/tex] seconds.
