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A moving company collects data about couches they move and displays the data in a table. The force along the ramp, the angle of the ramp, and the height the couch is raised from the ground to the platform is shown below.

A 4 column table with 3 rows. The first column is labeled couch with entries 1, 2, 3. The second column is labeled force in newtons with entries 450, 475 and 450. The third column is labeled angle of degrees above the horizontal with entries 25, 30 and 35. The last column is labeled height in meters with entries 12, 15 , 15.
Which statements correctly compare the work done to move the couches? Select three options.

The same amount of work is done for couches 1 and 3.
The same amount of work is done for couches 2 and 3.
The work for couch 1 is greater than the work for couch 2.
The work for couch 2 is less than the work for couch 3.
The work for couch 3 is greater than the work for couch 1.
The work for couch 2 is greater than the work for couch 1.

1 answer

To compare the work done to move the couches, we can use the formula for work done, which is given by:

\[ \text{Work} = \text{Force} \times \text{Distance} \times \cos(\theta) \]

where:

  • \( \text{Force} \) is the force applied,
  • \( \text{Distance} \) is the distance moved in the direction of the force (which can be inferred using the height and angle of the ramp),
  • \( \theta \) is the angle of the ramp above the horizontal.

However, for our purposes, we know that the height can also help in estimating how much work is done since the height the couch is raised will directly reflect on the work done against gravity.

Given Data from the Table:

  • Couch 1: Force = 450 N, Height = 12 m
  • Couch 2: Force = 475 N, Height = 15 m
  • Couch 3: Force = 450 N, Height = 15 m

Calculation of Work Done:

  1. Work for Couch 1: \[ \text{Work}_1 = 450 , \text{N} \times 12 , \text{m} = 5400 , \text{J} \]

  2. Work for Couch 2: \[ \text{Work}_2 = 475 , \text{N} \times 15 , \text{m} = 7125 , \text{J} \]

  3. Work for Couch 3: \[ \text{Work}_3 = 450 , \text{N} \times 15 , \text{m} = 6750 , \text{J} \]

Comparison:

  • Work for Couch 1 (5400 J) vs Couch 3 (6750 J): Couch 1 does less work than Couch 3.
  • Work for Couch 2 (7125 J) vs Couch 3 (6750 J): Couch 2 does more work than Couch 3.
  • Work for Couch 2 (7125 J) vs Couch 1 (5400 J): Couch 2 does more work than Couch 1.

Correct Statements:

Using the calculations above, the following statements are correct:

  1. The same amount of work is done for couches 1 and 3. (Incorrect)
  2. The same amount of work is done for couches 2 and 3. (Incorrect)
  3. The work for couch 1 is greater than the work for couch 2. (Incorrect)
  4. The work for couch 2 is less than the work for couch 3. (Incorrect)
  5. The work for couch 3 is greater than the work for couch 1. (Correct)
  6. The work for couch 2 is greater than the work for couch 1. (Correct)

Final Answers:

So the correct options are:

  • The work for couch 3 is greater than the work for couch 1.
  • The work for couch 2 is greater than the work for couch 1.
  • The work for couch 2 is greater than the work for couch 3. (Incorrect, so not selected)

Thus, the final three correct comparisons based on the problem context are:

  1. The work for couch 3 is greater than the work for couch 1.
  2. The work for couch 2 is greater than the work for couch 1.
  3. The work for couch 2 is greater than the work for couch 3. (Inferred)

However, only two statements are directly verifiable from the comparisons.