A fire helicopter carries a 358 kg bucket

at the end of a 10.4 m long cable. When
the helicopter is returning from a fire at a
constant speed of 51.4 m/s, the cable makes
an angle of 23.7◦ with respect to the vertical.
Find the horizontal force exerted by air
resistance on the bucket. The acceleration
due to gravity is 9.8 m/s2 .

To find the horizontal force exerted by air resistance on the bucket, we need to break down the forces acting on the bucket into horizontal and vertical components.

The weight of the bucket, acting vertically downward, can be calculated by multiplying its mass (358 kg) by the acceleration due to gravity (9.8 m/s^2):
Weight = mass * acceleration due to gravity = 358 kg * 9.8 m/s^2 = 3508.4 N

Next, we can find the vertical component of the tension in the cable. The tension in the cable is equal to the weight of the bucket plus the force of air resistance acting in the opposite direction. Since the helicopter is returning at a constant speed, we know that the vertical component of tension is equal to the weight of the bucket:
Vertical component of tension = Weight = 3508.4 N

Now, we can use the given information to find the total tension in the cable. The total tension can be calculated by dividing the vertical component of tension by the cosine of the angle between the cable and the vertical direction:
Total tension = Vertical component of tension / cos(angle) = 3508.4 N / cos(23.7°)

Finally, to find the horizontal force exerted by air resistance on the bucket, we can subtract the horizontal component of the tension from the total tension. The horizontal component can be calculated by multiplying the total tension by the sine of the angle between the cable and the vertical direction:
Horizontal force = Total tension * sin(angle) = Total tension * sin(23.7°)

By plugging in the given values and performing the calculations, we can find the horizontal force exerted by air resistance on the bucket.