Allegra's favorite ride at the Barrel-O-Fun Amusement Park is the Flying Umbrella, which is lifted by a hydraulic jack. The operator activates the ride by applying a forcce of 72 N to a 3.0-cm-wide cylindrical piston, which holds the 20000. N ride off the ground. What is the diameter of the piston that holds the ride?
Please Help!
To find the diameter of the piston, we can use the formula relating force, pressure, and area:
Pressure = Force / Area
The area of the piston can be calculated using the formula for the area of a circle:
Area = π * (radius)^2
Since the piston is cylindrical, the radius is half the diameter.
Let's solve it step by step:
1. Convert the width of the piston from centimeters to meters:
Width = 3.0 cm = 0.03 m
2. Calculate the area of the piston using the formula for the area of a circle:
Area = π * (radius)^2
3. Determine the radius using the width (diameter) of the piston:
Diameter = Width
Radius = Diameter / 2
4. Substitute the values into the formula and solve for the area of the piston:
Area = π * (0.03 m / 2)^2
5. Calculate the force on the piston using the formula for pressure:
Pressure = Force / Area
6. Substitute the known values of force and area into the formula and solve for pressure:
Pressure = 7200 N / Area
7. Calculate the pressure exerted on the hydraulic piston:
Pressure = 20000 N / Area
8. Equate the two pressure equations and solve for the area:
7200 N / Area = 20000 N / Area
9. Cross-multiply and solve for Area:
7200 N * Area = 20000 N * Area
10. Cancel out the area and solve for Area:
7200 N = 20000 N
11. Divide both sides of the equation by 7200 N:
Area = 20000 N / 7200 N
12. Calculate the numerical value of Area:
Area = 2.777 m^2
13. Substitute the value of Area into the formula for the area of a circle and solve for the radius:
Area = π * (radius)^2
14. Solve for radius:
2.777 m^2 = π * (radius)^2
15. Divide both sides of the equation by π to isolate (radius)^2:
(radius)^2 = 2.777 m^2 / π
16. Take the square root of both sides of the equation to solve for the radius:
radius = √(2.777 m^2 / π)
17. Calculate the numerical value of the radius:
radius ≈ 1.06 m
18. Finally, calculate the diameter using the formula for the diameter of a circle:
Diameter = 2 * radius
Diameter ≈ 2 * 1.06 m
Diameter ≈ 2.12 m
Therefore, the diameter of the piston that holds the ride is approximately 2.12 meters.
To find the diameter of the piston that holds the ride, we can use the formula for pressure, which is force divided by the area:
Pressure = Force / Area
In this case, the force applied by the operator is 72 N, and the width of the piston is given as 3.0 cm. However, we need to convert the width to meters for the units to be consistent. Since there are 100 centimeters in a meter, we can convert the width to meters by dividing it by 100:
Width (m) = Width (cm) / 100 = 3.0 cm / 100 = 0.03 m
Now, we can calculate the area of the piston:
Area = Width × Length
We are given the force holding the ride off the ground as 20,000 N. To calculate the length of the piston, we can rearrange the formula for pressure:
Pressure = Force / Area
Pressure = 20,000 N / Area
Since we know the pressure exerted by the operator is the same as the pressure needed to hold the ride off the ground, we can set up an equation:
72 N / (0.03 m × Length) = 20,000 N / Area
Dividing both sides of the equation by 72 N gives:
1 / (0.03 m × Length) = 20,000 N / (72 N × Area)
Simplifying further:
1 / (0.03 m × Length) = 277.78 / Area
To find the area, we can use the formula for the area of a circle:
Area = π × (Radius)^2
However, we need to find the radius first, which is half the diameter of the piston. Let's represent the radius as r and the diameter as d:
d = 2 × r
Now we can substitute the diameter into the equation for the area and rearrange for the radius:
Area = π × ((d/2)^2)
Area = π × r^2
r^2 = Area / π
r = √(Area / π)
Substituting the area we found in previous steps, we have:
r = √(277.78 / Area)
Now, we can substitute the formula for the area into the equation for the pressure:
1 / (0.03 m × Length) = 277.78 / (π × r^2)
1 / (0.03 m × Length) = 277.78 / (π × (√(277.78 / Area))^2)
1 / (0.03 m × Length) = 277.78 / (π × (277.78 / Area))
1 / (0.03 m × Length) = Area / π
Rearranging the equation:
Length = Area × (0.03m / π)
Finally, we can substitute the known values into the equation to find the length:
Length = (72 N / 277.78) × (0.03 m / π)
Calculating the length gives:
Length ≈ 0.000774 m
Now that we have the length of the piston, we can find the diameter by doubling the radius:
Diameter = 2 × r
Finally, substituting the values we got:
Diameter = 2 × √(Area / π)
Diameter ≈ 2 × √(277.78 / Area)
Diameter ≈ 2 × √(277.78 / (72 N / 277.78))
Calculating the diameter gives the answer to the question.