To determine which graph represents the leaf’s height as a function of time after it falls from a height of 12 feet, we can start by calculating the height of the leaf at different intervals of time.
The leaf falls from a height of 12 feet at a rate of 0.25 feet per second. We can use the formula for height \( H(t) \) after \( t \) seconds:
\[ H(t) = 12 - 0.25t \]
Now, we can calculate a few height values for various time intervals:
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At \( t = 0 \) seconds: \[ H(0) = 12 - 0.25(0) = 12 \text{ feet} \]
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At \( t = 4 \) seconds: \[ H(4) = 12 - 0.25(4) = 12 - 1 = 11 \text{ feet} \]
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At \( t = 8 \) seconds: \[ H(8) = 12 - 0.25(8) = 12 - 2 = 10 \text{ feet} \]
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At \( t = 12 \) seconds: \[ H(12) = 12 - 0.25(12) = 12 - 3 = 9 \text{ feet} \]
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At \( t = 16 \) seconds: \[ H(16) = 12 - 0.25(16) = 12 - 4 = 8 \text{ feet} \]
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At \( t = 20 \) seconds: \[ H(20) = 12 - 0.25(20) = 12 - 5 = 7 \text{ feet} \]
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At \( t = 24 \) seconds: \[ H(24) = 12 - 0.25(24) = 12 - 6 = 6 \text{ feet} \]
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Finally, it will hit the ground (0 feet) at: \[ 0 = 12 - 0.25t \implies 0.25t = 12 \implies t = 48 \text{ seconds} \]
Now let's analyze the given options: We need to find a graph where the line starts at 12 feet (at time \( t = 0 \)), is downward sloping, and reaches 0 feet when \( t = 48 \) seconds.
- The first graph starts at \( (4, 20) \), which cannot be correct since the height must start at 12 feet.
- The second graph starts at \( (0, 12) \) but does not go down to 0 feet in 48 seconds.
- The third graph starts at \( (0, 16) \), which cannot be correct since the height must start at 12 feet.
- The fourth graph starts at \( (0, 12) \) but still does not go down to 0 feet.
None of the options appear to reach 0 feet by 48 seconds as calculated. However, based on the trend of height decreasing over time, the graph that starts at \( (0, 12) \) would be the most reasonable since the starting height is correct.
Thus, the graph that likely represents the leaf’s height as a function of time is the second one: A coordinate plane showing Falling Leaf, Time in seconds on the x-axis and Height in feet on the y-axis. A line is passing through (0, 12), (8, 8), and (20, 2).
This is the only one that begins at 12 feet. While it does not account for the exact timing of reaching 0 feet at 48 seconds, it is the most consistent with the available data.
2 answers