A bucket begins weighing 15 pounds, including the sand it holds. The bucket is to be lifted to the top of a 50 foot
tall building by a rope of negligible weight. However, the bucket has a hole in it, and leaks 0.1 pounds of sand
each foot it is lifted. Find the work done lifting the bucket to the top of the building.
Thanks sir
To find the work done lifting the bucket to the top of the building, we need to calculate the total weight of the bucket at each foot interval and multiply it by the height it is being lifted.
Given:
Initial weight of the bucket with sand = 15 pounds
Leakage rate = 0.1 pounds/foot
Height of the building = 50 feet
Let's calculate the weight of the bucket at each foot interval:
At 1 foot:
Weight of the bucket = Initial weight - (Leakage rate × 1 foot)
Weight at 1 foot = 15 - (0.1 × 1) = 14.9 pounds
At 2 feet:
Weight of the bucket = Initial weight - (Leakage rate × 2 feet)
Weight at 2 feet = 15 - (0.1 × 2) = 14.8 pounds
Similarly, we can calculate the weight at each foot interval up to 50 feet.
Now, we can calculate the work done by multiplying the weight at each foot interval by the height:
Work done = (Weight at 1 foot × 1 foot) + (Weight at 2 feet × 1 foot) + ... + (Weight at 50 feet × 1 foot)
Let's calculate the work done:
Work done = (14.9 × 1) + (14.8 × 1) + ... + (weight at 50 feet × 1)
Since the weight changes uniformly with each foot interval, we can calculate the average weight and multiply it by the height:
Average weight = (15 + weight at 50 feet) / 2
Work done = Average weight × Height
Now, let's calculate the average weight:
Average weight = (15 + weight at 50 feet) / 2
To find the weight at 50 feet, we need to consider the leakage rate for each foot:
Weight at 50 feet = Initial weight - (Leakage rate × 50)
Weight at 50 feet = 15 - (0.1 × 50) = 15 - 5 = 10 pounds
Substituting the values, we get:
Average weight = (15 + 10) / 2 = 12.5 pounds
Work done = Average weight × Height = 12.5 × 50 = 625 foot-pounds
Therefore, the work done lifting the bucket to the top of the building is 625 foot-pounds.
To find the work done lifting the bucket to the top of the building, we can use the concept of work done against gravity.
Work done against gravity is calculated by multiplying the force applied in the direction of displacement by the distance over which the force is applied. In this case, the force is the weight of the bucket.
Initially, the weight of the bucket is 15 pounds, including the sand it holds. However, it is important to consider that the weight is decreasing as the sand leaks out of the bucket. We need to find the total weight of the bucket at each incremental distance it is lifted.
Let's break down the problem into smaller steps:
Step 1: Calculate the weight of the sand leaked at each foot
The bucket loses 0.1 pounds of sand per foot it is lifted. Since we are considering the incremental distance, we need to multiply the weight loss per foot by the distance lifted. Let's call the distance lifted as "x" (in feet).
Weight of sand leaked (in pounds) = 0.1 * x
Step 2: Calculate the weight of the bucket at each foot
The weight of the bucket at each foot is the initial weight minus the weight of the sand leaked.
Weight of the bucket (in pounds) = 15 - (0.1 * x)
Step 3: Calculate the work done at each foot
The work done at each foot is the product of the weight of the bucket and the incremental distance lifted.
Work done at each foot (in foot-pounds) = (Weight of the bucket) * (incremental distance) = (15 - 0.1x) * 1
Step 4: Integrate to find the total work done
Now, we need to integrate the equation for the work done at each foot over the range from 0 to 50 feet to find the total work done.
Total work done (in foot-pounds) = ∫[(15 - 0.1x) * 1] dx, where x ranges from 0 to 50
Solving this integral will give us the work done lifting the bucket to the top of the building.
weight at top = 15 -.1*50
= 10
average weight = (15+10)/2 = 12.5
12.5 * 50 = 625 ft pounds