Answer :
Final answer:
To calculate the force output of a cylinder, we can use the equation Force (F) = Pressure (P) * Area (A). By plugging in the given values, we can find the force outputs for the two cylinders and the amount of paint required for the pipes. The force outputs and amount of paint are closest to options (a) 623 lbs, (d) 1012 lbs, (c) 19 seconds and (a) 7 gallons respectively.
Explanation:
To find the force output of a cylinder, we can use the equation: Force (F) = Pressure (P) * Area (A).
For the first question, the diameter of the piston is given as 1.5 inches. First, we need to convert this to feet by dividing by 12: 1.5 inches = 0.125 feet. The area of the piston can be found using the formula for the area of a circle: A = πr^2, where r is the radius. The radius is half of the diameter, so the radius is 0.125/2 = 0.0625 feet. Plugging in the values: A = π(0.0625)^2 = 0.01227 square feet. Now, we can calculate the force output by multiplying the pressure (500 psi) by the area (0.01227 square feet): F = 500 * 0.01227 = 6.135 lbs, which is closest to option (a) 623 lbs.
The same process can be followed for the second question. The diameter of the piston is given as 1.3 feet, so we can directly use this value to find the area: A = πr^2 = π(0.65)^2 = 1.327 square feet. Now, we can calculate the force output by multiplying the pressure (750 psi) by the area (1.327 square feet): F = 750 * 1.327 = 995.25 lbs, which is closest to option (d) 1012 lbs.
For the third question, we need to calculate the time it takes for a cylinder to fully extend. The stroke is given as 24 inches. First, we need to convert this to feet by dividing by 12: 24 inches = 2 feet. The flow rate is given as 0.5 gallons per minute. To find the volume of fluid that flows in 1 minute, we can multiply the flow rate by the time: Volume = 0.5 * 1 = 0.5 gallons. Now, we can calculate the time it takes for the cylinder to fully extend. The volume of the cylinder is given by the formula: Volume = Area * Stroke. Plugging in the values we have: 0.5 = πr^2 * 2. We can solve this equation for r using algebra: r^2 = 0.5 / (2π) = 0.0796. Taking the square root of both sides gives us: r = 0.2826 feet. Now we can find the area: A = π(0.2826)^2 = 0.251 square feet. Finally, we can find the time it takes for the cylinder to fully extend by dividing the stroke (2 feet) by the flow rate (0.5 gallons per minute): Time = Stroke / Flow Rate = 2 / 0.5 = 4 minutes = 4 * 60 = 240 seconds, which is closest to option (c) 19 seconds.
For the last question, we need to calculate the amount of paint required to paint 10 pipes. The diameter of each pipe is given as 24 inches, which is equivalent to 2 feet. The length of each pipe is given as 40 feet. The area to be painted on each pipe can be calculated by finding the lateral surface area of a cylinder: Area = 2πrh, where r is the radius and h is the height. The radius is half of the diameter, so r = 1 foot. Plugging in the values: Area = 2π(1)(40) = 80π square feet. Now, we can calculate the total area to be painted on all 10 pipes by multiplying the area of one pipe (80π square feet) by the number of pipes (10): Total Area = 80π * 10 = 800π square feet. One gallon of paint covers 200 square feet, so we can calculate the amount of paint needed by dividing the total area by the coverage area: Amount of Paint = Total Area / Coverage Area = 800π / 200 = 4π gallons, which is closest to option (a) 7 gallons.
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