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The heights of two elevators can be modelled by linear functions. At time t = 0, Elevator A is 16 feet above ground floor. It descends at a rate of 4 feet per minute. The height of Elevator B is shown in the graph. Select all true statements.

Elevator B graph of a diagonal line on a coordinate plane going down and to the right with Time in minutes on the x axis and height in feet on the y axis. The line begins at the point zero comma 16 and passes through the point 2 comma 10.

The initial height of Elevator A is less than the initial height of Elevator B.


Elevator A descends at a faster rate than Elevator B.


Elevator A will reach the ground first.


Elevator B will reach the ground first.


The initial height of Elevator B is less than the initial height of Elevator A.

1 answer

To analyze the situation, let's summarize the information given about both elevators.

Elevator A:

  • Initial height at \( t = 0 \) is \( 16 \) feet.
  • Descend rate is \( 4 \) feet per minute.

The height of Elevator A as a function of time \( t \) can be expressed as: \[ h_A(t) = 16 - 4t \]

Elevator B:

  • Initial height at \( t = 0 \) is also \( 16 \) feet (given that it starts at the point \( (0, 16) \)).
  • The graph indicates that at \( t = 2 \) minutes, the height is \( 10 \) feet. This means it descends \( 6 \) feet in \( 2 \) minutes, which yields a descent rate of: \[ \text{Rate} = \frac{10 - 16}{2} = -3 \text{ feet per minute} \]

The height of Elevator B can be expressed as: \[ h_B(t) = 16 - 3t \]

Now, let's evaluate the statements.

  1. The initial height of Elevator A is less than the initial height of Elevator B.

    • False. Both heights are \( 16 \) feet at \( t = 0 \).

  2. Elevator A descends at a faster rate than Elevator B.

    • True. Elevator A descends at \( 4 \) feet per minute, while Elevator B descends at \( 3 \) feet per minute.

  3. Elevator A will reach the ground first.

    • True. To determine when both elevators reach the ground:
      • For Elevator A: \( 0 = 16 - 4t \rightarrow 4t = 16 \rightarrow t = 4 \) minutes.
      • For Elevator B: \( 0 = 16 - 3t \rightarrow 3t = 16 \rightarrow t \approx 5.33 \) minutes.
    • So, Elevator A reaches the ground first.

  4. Elevator B will reach the ground first.

    • False. Based on the calculations above, Elevator B takes longer to reach the ground.

  5. The initial height of Elevator B is less than the initial height of Elevator A.

    • False. As noted previously, both elevators start at \( 16 \) feet.

Now, summarizing the true statements, we have:

  • Elevator A descends at a faster rate than Elevator B.
  • Elevator A will reach the ground first.

Thus, the true statements are:

  • Elevator A descends at a faster rate than Elevator B.
  • Elevator A will reach the ground first.