Final answer:
The total work done on the vacuum is calculated by finding the horizontal component of the force exerted by the custodian and subtracting the work done against friction. The work done by the custodian's force is approximately 262.575 J, and the work done against friction is 191.7 J, resulting, in a total work of approximately 70.875 J.
Step-by-step explanation:
To find the total work done on the vacuum, we need to consider both the force exerted by the custodian and the force of friction. The work done by the custodian can be calculated by breaking down the 33.9 N force into its horizontal component that does work against friction and using the formula work (W) = force (F) × distance (d) × cos(θ), where θ is the angle of force applied. The formula accounts for the direction of the force since only the component of the force that acts in the direction of movement does work.
The horizontal component of the force is Fx = F × cos(θ), or 33.9 N × cos(55°), which equals approximately 19.45 N. The work done by this component against friction over 13.5 meters is W = 19.45 N × 13.5 m, which equals approximately 262.575 J.
However, since there's also a friction force that opposes the motion, we need to subtract the work done by the friction force. The work done against friction is Wf = friction force (Ff) × distance (d), which is 14.2 N × 13.5 m equals 191.7 J.
Therefore, the total work done on the vacuum is the work done by the custodian's force minus the work done by friction: 262.575 J - 191.7 J equals approximately 70.875 J.