Calculus 2. Tom and Mike have a bet as to who will do the most work today. Mike has to compress a coil 200 feet. It takes Mike 250 lbs to compress the coil 10 feet. Tom needs to pump water through the top of a cylindrical tank sitting on the ground. The tank is half full, has a radius of 4 feet, and is 10 feet tall. Who does the most work? Explain.
To determine who does the most work, we need to calculate the work done by both Mike and Tom.
First, let's calculate the work done by Mike. We know that it takes Mike 250 lbs of force to compress the coil 10 feet. So, the work done by Mike can be calculated using the formula:
Work = Force × Distance
In this case, the distance is 200 feet, and the force required to compress the coil 10 feet is 250 lbs. We can set up a proportion to find the force required to compress the coil 200 feet:
200 ft ÷ 10 ft = X lbs ÷ 250 lbs
Solving this proportion, we get:
X = (200 ft ÷ 10 ft) × 250 lbs
X = 5000 lbs
So, Mike needs to exert 5000 lbs of force to compress the coil 200 feet.
Now, let's calculate the work done by Tom. Tom needs to pump water through the top of a cylindrical tank. The work done in this case can be calculated as the change in potential energy of the water.
The potential energy of the water in the tank is given by the formula:
Potential Energy = m × g × h
Where m is the mass of the water, g is the acceleration due to gravity, and h is the height raised.
Since the tank is half full, the mass of the water can be calculated as the volume of the cylinder filled with water multiplied by the density of water. The volume of the cylinder is π × r^2 × h, and the density of water is approximately 62.4 lbs/ft^3.
Substituting the values into the formula, we get:
Potential Energy = (π × r^2 × h) × 62.4 lbs/ft^3 × g × h
= π × (4 ft)^2 × (10 ft) × 62.4 lbs/ft^3 × g × (10 ft)
= 5024π lbs × g ft
Since the height through which the water is raised is 10 feet, the potential energy simplifies to:
Potential Energy = 50240π lbs × g ft
Now, since we want to compare the work done by Mike and Tom, we need to convert the potential energy to work done. Work is calculated as the product of force and distance traveled.
In this case, the distance traveled is the same for both Mike and Tom, which is 10 feet. So, we can calculate the work done by Tom as:
Work (Tom) = Force (Tom) × Distance
= Potential Energy × (1/10) [Dividing by 10 to convert ft to 1 ft]
Substituting the value of potential energy, we get:
Work (Tom) = 50240π lbs × g ft × (1/10) = 5024π lbs × g ft
Now, to compare the work done by Mike and Tom, we need to compare the values of 5000 lbs and 5024π lbs × g ft.
Note that the value of g, the acceleration due to gravity, is approximately 32.17 ft/s^2.
Thus, comparing the two values, we can determine who does the most work.