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.
To find the work done lifting the bucket to the top of the building, we can use the concept of work done against gravity.
The work done is given by the equation:
Work = Force x Distance
The force is the weight of the bucket, which changes as sand leaks out. The weight can be calculated by subtracting the weight of the leaked sand from the initial weight of the bucket.
Let's calculate the work done step by step:
Step 1: Calculate the weight of the bucket with sand at the base.
Weight of the bucket (initial) = 15 pounds
Step 2: Calculate the weight of the bucket at each foot it is lifted.
Weight of the bucket (at any height h) = Weight of the bucket (initial) - (0.1 pounds/foot x h)
Step 3: Calculate the work done at each foot.
Work done at each foot (h) = Weight of the bucket (at any height h) x h
Step 4: Calculate the total work done to lift the bucket to the top of the building.
Total work done = Sum of the work done at each foot (from 0 to 50 feet)
Let's calculate the work done step by step:
Step 1: Weight of the bucket (initial) = 15 pounds
Step 2: Weight of the bucket (at any height h) = 15 - (0.1 x h)
Step 3: Work done at each foot (h) = (15 - 0.1h) x h
Step 4: Total work done = ∑ [(15 - 0.1h) x h] from h = 0 to h = 50
Using the formula for the sum of arithmetic series, the total work done can be calculated as:
Total work done = (n/2) x [a + l]
Where:
n is the number of terms (in this case, 50)
a is the first term (h = 0)
l is the last term (h = 50)
Plugging in the values:
Total work done = (50/2) x [(15 - 0.1(0)) + (15 - 0.1(50))]
Simplifying:
Total work done = 25 x [15 + 15 - 5]
Total work done = 25 x 25
Total work done = 625 foot-pounds
Therefore, the work done to lift 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 need to determine the total weight of sand that is lifted.
First, let's find the weight of the sand after each foot it is lifted. Since the bucket leaks 0.1 pounds of sand for each foot, we can calculate the weight of sand at each foot using the formula:
Weight of sand = Initial weight of sand - (Leakage rate × Distance lifted)
Given that the initial weight of the bucket (with sand) is 15 pounds, and the rope is lifted 50 feet, we can calculate the weight of sand at the top of each foot:
At 1 foot: Weight of sand at 1 foot = 15 - (0.1 × 1) = 14.9 pounds
At 2 feet: Weight of sand at 2 feet = 15 - (0.1 × 2) = 14.8 pounds
...
At 50 feet: Weight of sand at 50 feet = 15 - (0.1 × 50) = 10 pounds
Now, we need to determine the average weight of the sand while it is being lifted. We can do this by finding the average of the weight of sand at each foot.
Average weight of sand = (Weight of sand at 1 foot + Weight of sand at 2 feet + ... + Weight of sand at 50 feet) / Number of feet
To find the sum of the weight of sand at each foot, we can apply the arithmetic series formula:
Sum = (First term + Last term) × Number of terms / 2
First term = 14.9 pounds, Last term = 10 pounds, Number of terms = 50
Sum = (14.9 + 10) × 50 / 2 = 24.9 × 25 = 622.5 pounds
Average weight of sand = 622.5 / 50 = 12.45 pounds
Finally, we can calculate the work done by multiplying the average weight of sand by the distance lifted:
Work done = Average weight of sand × Distance lifted
Work done = 12.45 × 50 = 622.5 foot-pounds
Therefore, the work done in lifting the bucket to the top of the building is 622.5 foot-pounds.