A mountain climber is about to haul up a 30 -m length of hanging rope. How much work will it take if the rope weighs \( 0.75 \mathrm{~N} / \mathrm{m} \) ? The amount of work required is \( \square \mathrm{J} \). (Type an integer or a decimal.)

To calculate the work done in hauling up the rope, we can use the formula for work done against gravity, which is given by: \[ W = F \times d \] where \( F \) is the force (weight of the rope), and \( d \) is the distance the rope is raised. First, we need to find the total weight of the whole rope. Since the rope weighs \( 0.75 \mathrm{~N/m} \) and its length is \( 30 \, \mathrm{m} \): \[ \text{Total Weight} = \text{Weight per meter} \times \text{Length} = 0.75 \, \mathrm{N/m} \times 30 \, \mathrm{m} = 22.5 \, \mathrm{N} \] Next, we want to consider that as the climber hauls the rope, the average force needed is half of the total weight, since the entire 30 m of rope is not lifted all at once. Therefore: \[ \text{Average Force} = \frac{22.5 \, \mathrm{N}}{2} = 11.25 \, \mathrm{N} \] The distance \( d \) is simply the length of the rope being lifted, which is \( 30 \, \mathrm{m} \). Now we can find the work done: \[ W = F \times d = 11.25 \, \mathrm{N} \times 30 \, \mathrm{m} = 337.5 \, \mathrm{J} \] Thus, the amount of work required is \( 337.5 \, \mathrm{J} \).