The ski slopes at Bluebird Mountain make use of tow ropes to transport snowboarders and skiers to the summit of the hill. One of the tow ropes is powered by a 22-kW motor which pulls skiers along an icy incline of 14° at a constant speed. Suppose that 18 skiers with an average mass of 48 kg hold onto the rope and suppose that the motor operates at full power.

a. Determine the cumulative weight of all these skiers.
b. Determine the force required to pull this amount of weight up a 14° incline at a constant speed.
c. Determine the speed at which the skiers will ascend the hill.

A. mass X gravity X 18 people = 8476kgm/s^2

B. mass X gravity X sin(14) X 18 = 2050N

C. Power = Force X Velocity
22,000 = 2050 X Velocity
Velocity = 11m/s

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a. The cumulative weight of all the skiers can be calculated by multiplying the average mass of each skier by the total number of skiers:

Cumulative weight = average mass of each skier * total number of skiers
Cumulative weight = 48 kg * 18 skiers
Cumulative weight = 864 kg

b. The force required to pull this amount of weight up a 14° incline can be calculated using the formula:

Force = weight * sine(angle)
Force = 864 kg * sine(14°)
Force = 864 kg * 0.2425
Force ≈ 209.52 N

c. The speed at which the skiers will ascend the hill can be calculated using the formula:

Speed = Power / Force
Speed = 22 kW / 209.52 N
Speed ≈ 0.105 m/s

So, the skiers will ascend the hill at a speed of approximately 0.105 meters per second.

Let's break down each part of the problem and calculate the values step by step:

a. To determine the cumulative weight of all the skiers, we need to multiply the average mass of each skier (48 kg) by the number of skiers (18):

Cumulative weight = Average mass of each skier * Number of skiers
Cumulative weight = 48 kg * 18
Cumulative weight = 864 kg

Therefore, the cumulative weight of all the skiers is 864 kg.

b. To determine the force required to pull this amount of weight up a 14° incline at a constant speed, we need to consider the force of gravity acting on the skiers. The force of gravity is given by the formula:

Force of gravity = Mass * gravitational acceleration

Gravitational acceleration is approximately 9.8 m/s².

Force of gravity = Cumulative weight * gravitational acceleration
Force of gravity = 864 kg * 9.8 m/s²
Force of gravity = 8476.8 N

Now, we need to consider the force required to pull this weight up the incline. The force required can be calculated using the formula:

Force required = Force of gravity * sine(angle of incline)

Note: We use the sine function here because the weight is acting perpendicular to the incline, and the force required is acting along the incline.

Force required = 8476.8 N * sin(14°)
Force required = 8476.8 N * 0.2419
Force required = 2050.38 N

Therefore, the force required to pull this amount of weight up a 14° incline at a constant speed is approximately 2050.38 N.

c. To determine the speed at which the skiers will ascend the hill, we need to consider the power of the motor. Power is defined as the work done per unit time. In this case, the motor is doing work by pulling the skiers up the incline.

Power = Force * velocity

Given that the motor operates at full power (22 kW) and the force required is 2050.38 N, we can rearrange the formula to solve for velocity:

Velocity = Power / Force required

First, let's convert the power from kilowatts (kW) to watts (W):

Power = 22 kW * 1000
Power = 22000 W

Velocity = 22000 W / 2050.38 N
Velocity ≈ 10.73 m/s

Therefore, the speed at which the skiers will ascend the hill is approximately 10.73 m/s.

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