To determine the useful work done in pumping 1,000 gallons of water to a height of 60 feet, we first need to calculate the weight of the water and then the work done against gravity. After that, we can find the horsepower developed and the horsepower required based on the efficiency of the pumping system.
Step 1: Calculate the Weight of the Water
Water has a density of approximately 8.34 pounds per gallon. Therefore, the total weight of 1,000 gallons of water can be calculated as follows:
- Weight = Volume × Density
- Weight = 1,000 gallons × 8.34 pounds/gallon = 8,340 pounds
Step 2: Calculate the Work Done
The work done in lifting an object against gravity is given by the formula:
Substituting the values we have:
- Work = 8,340 pounds × 60 feet = 500,400 foot-pounds
Step 3: Determine Horsepower Developed
Horsepower is a measure of power, which is the rate at which work is done. The formula for horsepower is:
First, we need to convert the time from minutes to seconds since horsepower is typically expressed in terms of seconds:
- Time = 25 minutes × 60 seconds/minute = 1,500 seconds
Now we can calculate the horsepower developed:
- Horsepower = 500,400 foot-pounds / 1,500 seconds ≈ 333.6 horsepower
Step 4: Adjust for Efficiency
Given that the efficiency of the pumping arrangement is 55%, we need to find the horsepower that must be supplied to achieve this output. The formula to find the required horsepower based on efficiency is:
- Required Horsepower = Horsepower Developed / Efficiency
Substituting the values:
- Required Horsepower = 333.6 horsepower / 0.55 ≈ 606.5 horsepower
Summary of Results
In summary, the useful work done in pumping 1,000 gallons of water to a height of 60 feet is 500,400 foot-pounds. The horsepower developed during this process is approximately 333.6 horsepower, and considering the efficiency of the pumping system, the horsepower that must be supplied is about 606.5 horsepower.
