The function \( d = |0.75t - 4| \) represents the distance in floors from the 4th floor as a function of time \( t \). To understand the key features of this function, we can analyze it:
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Initial Distance: When \( t = 0 \) (at the initial moment when the pulley leaves the ground), we have: \[ d = |0.75(0) - 4| = |-4| = 4. \] This means the pulley starts 4 floors below the 4th floor.
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Rate of Movement: The term \( 0.75t \) indicates that the pulley ascends at a rate of 0.75 floors per second. This means that, as time increases, the distance to the 4th floor will change based on how quickly the pulley moves.
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Critical Point: The function inside the absolute value, \( 0.75t - 4 \), will become zero when \( t = \frac{4}{0.75} = \frac{16}{3} \approx 5.33 \) seconds. This point corresponds to the moment when the pulley reaches the 4th floor, and the distance \( d \) becomes zero.
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Behavior After Reaching the 4th Floor: After \( t \approx 5.33 \) seconds, the function will start to describe the distance above the 4th floor (as the value inside the absolute value becomes positive), and \( d \) will increase linearly with a slope of 0.75.
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V-Shape Graph: The graph of this function will resemble a "V" shape, with the vertex at the point where \( t \approx 5.33 \) seconds (the 4th floor), and it opens upwards because of the absolute value.
Overall, the key features of the function reflect the initial floor distance, the rate of ascent, the critical point where the pulley reaches the 4th floor, and the linear behavior of the distance from that point onwards.