A bucket begins weighing 30 pounds, including the sand it holds. The bucket is to be lifted to the top of a 25 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.
work = force * distance, so
∫[0,25] 30-.1x dx
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 every foot it is lifted, taking into account the loss of sand through the hole.
We know that the bucket begins weighing 30 pounds, including the sand it holds. Since it loses 0.1 pounds of sand per foot, the weight of the bucket at each foot can be calculated as follows:
1st foot: 30 - 0.1 = 29.9 pounds
2nd foot: 29.9 - 0.1 = 29.8 pounds
3rd foot: 29.8 - 0.1 = 29.7 pounds
...
25th foot: 30 - (0.1 x 25) = 27.5 pounds
To find the work done, we need to calculate the total weight lifted by summing up the weight at each foot. We can do this by considering the average weight at each foot. Since the weight changes linearly, the average weight can be found by averaging the weight at the start and end of the distance:
Average weight = (initial weight + final weight) / 2
Average weight = (30 + 27.5) / 2 = 28.75 pounds
Now we can find the work done using the formula:
Work = force x distance
The force is equal to the average weight, and the distance is 25 feet:
Work = 28.75 pounds x 25 feet = 718.75 foot-pounds
Therefore, the work done lifting the bucket to the top of the building is 718.75 foot-pounds.
To find the work done lifting the bucket to the top of the building, we need to calculate the total amount of sand that the bucket loses and the force required to lift the bucket.
First, let's calculate how much sand the bucket will lose as it is lifted. The bucket loses 0.1 pounds of sand per foot it is lifted. The height of the building is 25 feet, so the total amount of sand lost during the ascent would be 0.1 pounds/foot * 25 feet = 2.5 pounds.
Now, let's calculate the force required to lift the bucket. The weight of the bucket, including the sand it holds, is initially 30 pounds. However, since the bucket loses 0.1 pounds of sand per foot, the weight of the bucket decreases as it is lifted. So, at a height of 25 feet, the weight of the bucket would be 30 pounds - 2.5 pounds = 27.5 pounds.
To calculate the work done, we use the formula: Work = force * distance. In this case, the force required to lift the bucket is 27.5 pounds, and the distance it is lifted is 25 feet. So, the work done lifting the bucket to the top of the building is: Work = 27.5 pounds * 25 feet = 687.5 foot-pounds.
Therefore, the work done lifting the bucket to the top of the building is 687.5 foot-pounds.