The function d=|0.75t−4|

Question

can be used to determine the distance in floors a pulley is from the 4th floor, where t is the time in seconds since the pulley left the ground floor.

What do the key features of the function reveal?

(1 point)
Responses

The function is a decreasing function and an increasing function, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases at a constant rate, and as the pulley travels away from the 4th floor, the distance from the 4th floor increases at a constant rate.
The function is a decreasing function and an increasing function, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases at a constant rate, and as the pulley travels away from the 4th floor, the distance from the 4th floor increases at a constant rate.

The function is neither increasing nor decreasing, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases but not at a constant rate and as the pulley travels away from the 4th floor, the distance from the 4th floor increases but not at a constant rate.
The function is neither increasing nor decreasing, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases but not at a constant rate and as the pulley travels away from the 4th floor, the distance from the 4th floor increases but not at a constant rate.

The function is an increasing function, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases at a constant rate.
The function is an increasing function, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases at a constant rate.

The function is a decreasing function, so as the pulley travels away from the 4th floor, the distance from the 4th floor increases at a constant rate.

Answer 1

Let's analyze the function \( d = |0.75t - 4| \).

1. Understanding the Function:

   • The function consists of the absolute value of a linear expression \( 0.75t - 4 \).
   • The expression \( 0.75t - 4 = 0 \) when \( t = \frac{4}{0.75} = \frac{16}{3} \approx 5.33 \) seconds. This means at \( t \approx 5.33 \) seconds, the pulley is exactly at the 4th floor (distance \( d = 0 \)).
   • For \( 0 \leq t < \frac{16}{3} \), \( d = 4 - 0.75t \) (a decreasing function).
   • For \( t > \frac{16}{3} \), \( d = 0.75t - 4 \) (an increasing function).

2. Key Features:

   • From \( t = 0 \) to \( t \approx 5.33 \), the distance \( d \) decreases (the pulley is moving up towards the 4th floor).
   • After \( t \approx 5.33 \), the distance \( d \) increases (the pulley is moving up and away from the 4th floor).

Given this analysis, the correct interpretation of the function includes both increasing and decreasing behavior, with a specific point at which the distance to the 4th floor is minimal (zero).

Correct Response: The function is a decreasing function and an increasing function, so as the pulley travels from the ground floor to the 4th floor, the distance from the 4th floor decreases at a constant rate, and as the pulley travels away from the 4th floor, the distance from the 4th floor increases at a constant rate.