Final answer:
The work done by a 70.0 kg sailor climbing a 28.3 m rope ladder at a 45.0° angle with the mast is 13720 Joules.
Step-by-step explanation:
The student has asked how much work a 70.0 kg sailor does while climbing a 28.3 m rope ladder at a 45.0° angle with the mast. To solve for the work done, we need to consider that work (W) is equal to the force exerted on an object (in this case, the weight of the sailor) times the distance the object is moved in the direction of the force (here, the vertical height gained) and is given by the formula:
W = F × d × cos(\Theta)
Since the sailor is moving upward against gravity, the force involved is his weight (mass × gravitational acceleration, g=9.8 m/s²). The height h that the sailor ascends vertically can be found using the sine function, as h = L × sin(α), where L is the length of the ladder and α is the angle with respect to the vertical. The work done is:
W = (mass × g) × h = (70.0 kg × 9.8 m/s²) × (28.3 m × sin(45°))
Calculating the sine of 45° gives us √2/2, so this simplifies to:
W = (70.0 kg × 9.8 m/s²) × (28.3 m × √2/2)
W = (686 N) × (20.0 m)
W = 13720 J (Joules)
Therefore, the work done by the sailor while climbing the rope ladder is 13720 Joules.