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A droplet of water drips from the end of a stalactite high up on the roof of a cave. The height of the water droplet above the cave floor in meters can be modeled by the expression 22−4.9t222 - 4.9t^2, where tt is the time in seconds after the water droplet begins to fall.\newlineWhat does the quantity 4.9t24.9t^2 represent in the expression?\newlineChoices:\newline(A)the distance in meters the water droplet has fallen after tt seconds\newline(B)the time in seconds it takes for the water droplet to reach a height of tt meters\newline(C)the time in seconds it takes for the water droplet to fall tt meters\newline(D)the height in meters of the water droplet above the cave floor after tt seconds\newline

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Q. A droplet of water drips from the end of a stalactite high up on the roof of a cave. The height of the water droplet above the cave floor in meters can be modeled by the expression 22−4.9t222 - 4.9t^2, where tt is the time in seconds after the water droplet begins to fall.\newlineWhat does the quantity 4.9t24.9t^2 represent in the expression?\newlineChoices:\newline(A)the distance in meters the water droplet has fallen after tt seconds\newline(B)the time in seconds it takes for the water droplet to reach a height of tt meters\newline(C)the time in seconds it takes for the water droplet to fall tt meters\newline(D)the height in meters of the water droplet above the cave floor after tt seconds\newline
  1. Height Expression Analysis: The expression for the height of the water droplet is 22−4.9t222 - 4.9t^2. We know that the initial height is 2222 meters, so the term 4.9t24.9t^2 must represent the distance fallen after tt seconds, because as time increases, the height decreases.
  2. Derivation of 4.9t24.9t^2: The term 4.9t24.9t^2 is derived from the physics equation for the distance fallen under gravity, which is (1/2)gt2(1/2)gt^2, where gg is the acceleration due to gravity (9.8 m/s29.8 \, \text{m/s}^2). So, 4.94.9 is half of 9.89.8, which confirms that 4.9t24.9t^2 is the distance fallen.
  3. Confirmation of Distance Fallen: Since 4.9t24.9t^2 represents the distance fallen, it matches with choice (A) the distance in meters the water droplet has fallen after tt seconds.

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