High School

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A waterfall has a height of 900 feet. A pebble is thrown upward from the top of the falls with an initial velocity of 12 feet per second. The height, [tex]h[/tex], of the pebble after [tex]t[/tex] seconds is given by the equation [tex]h = -16t^2 + 12t + 900[/tex].

How long after the pebble is thrown will it hit the ground?

Answer :

A graphing calculator shows the time to be 7.884 seconds.

_____
The quadratic formula tells you the time is
.. t = (-12 -√(12^2 +4*16*900))/(-32) = (3/8)*(1 +√401) . . . seconds

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Rewritten by : Barada

Final answer:

To find when the pebble hits the ground, solve for time using the quadratic equation from the given height function, resulting in around 6 seconds.

Explanation:

To find how long the pebble takes to hit the ground, we need to find the time t when its height h becomes 0. This means we need to solve the equation for t when h = 0:

-16t² + 12t + 900 = 0

This is a quadratic equation, and we can solve it using various methods like factoring, the quadratic formula, or completing the square. Here, we'll use the quadratic formula:

t = (-b ± √(b² - 4ac)) / 2a

where a = -16, b = 12, and c = 900:

t = (-12 ± √(12² - 4 × -16 × 900)) / (2 × -16)

t = (-12 ± √(614400)) / -32

t ≈ (-12 ± 247.88) / -32

There are two possible solutions:

t1 ≈ 25.50 seconds

t2 ≈ -5.50 seconds (we can discard this since time cannot be negative)

The pebble will hit the ground approximately 6 seconds after it is thrown.