Irrigating the largest area of a field measuring 216 meters by 278 meters will produce the most profit. Watering the area exactly once will not over or under water any area of the field. This task can be accomplished by using engineering design and simple geometry. The goal is to have a wheel go around a field and stop in the exact same place on the wheel and field.
The circle the wheel travels around the field is the turn-circle-circumference. To stop in the exact same place, the wheel’s circumference must divide evenly into the turn-circle’s-circumference.
(Instead of using meters you can scale this problem down to centimeters by using a field size of 21.6 cm by 27.8 cm. This will fit on 1.5 sheets of paper.)

(How to find the answer for 1a & b) First: choose an axle length that goes from the center to the edge of the field. This radius you chose will be the radius of the circle made on the field by the irrigation system. Use this length to determine the circumference of the circle the wheel will travel around the field. This large circle is the turn-circle-circumference.
Use this equation: Circumference = 2(3.14)(Radius)

1a. Provide the filled out equation for determining the turn-circle circumference. (Filled out means put the numbers in the equation.)

1b. Provide the circumference of the turn circle in cm. (Write the answer you calculated for 1a above.)

(How to find the answer for 2a & b) Second: determine the circumference of a wheel that will divide evenly into the turn-circle-circumference.
Use the equation: Distance = Wheel Circumference X Rotations
Distance will be the turn-circle-circumference you calculated above. You get to pick the number of rotations to make a reasonably sized wheel.

2a. Provide the filled out equation for determining the wheel circumference. (Filled out means put the numbers in the equation.)

2b. Provide the circumference of the wheel in meters. (Write the answer you calculated for 2a above.)

(How to find the answer for 3a & b) Third: prove that your wheel is a multiple of the turn-circle-circumference.
Fill out this equation by putting the number of rotations and wheel circumference into it. Then solve for distance.
Use the equation: Distance = Wheel Circumference X Rotations

3a. Fill out an equation that proves your wheel circumference is a multiple of your turn-circle.

3b. What was the result of the calculation above. Does this number equal the turn-circle-circumference in 1b above?

Make a simplified model of a center-pivot irrigation system consisting of only a wheel and axle. The path that the wheel travels is the turn-circle circumference. 1. Draw a large turn circle on 1 or 1.5 sheets of paper. 2. Make an axle the radius of the turn circle. 3. -Put a disk of paper on the end of an axle for a wheel. (Use a disk shaped rather than cylinder shaped wheel.)
4. Make a wheel with a circumference that is a multiple of the turn-circle’s circumference. 5. To test this, a mark on the wheel and a mark on the turn-circle will line up before and after the wheel rolls exactly once around the turn-circle.

Irrigating the largest area of a field measuring 216 meters by 278 meters will produce the most profit. Watering the area exactly once will not over or under water any area of the field. This task can be accomplished by using engineering design and simple geometry. The goal is to have a wheel go around a field and stop in the exact same place on the wheel and field.

The circle the wheel travels around the field is the turn-circle-circumference. To stop in the exact same place, the wheel’s circumference must divide evenly into the turn-circle’s-circumference.

(Instead of using meters you can scale this problem down to centimeters by using a field size of 21.6 cm by 27.8 cm. This will fit on 1.5 sheets of paper.)

(How to find the answer for 1a & b) First: choose an axle length that goes from the center to the edge of the field. This radius you chose will be the radius of the circle made on the field by the irrigation system. Use this length to determine the circumference of the circle the wheel will travel around the field. This large circle is the turn-circle-circumference.

Use this equation: Circumference = 2(3.14)(Radius)

1a. Provide the filled out equation for determining the turn-circle circumference. (Filled out means put the numbers in the equation.)

1b. Provide the circumference of the turn circle in cm. (Write the answer you calculated for 1a above.)

(How to find the answer for 2a & b) Second: determine the circumference of a wheel that will divide evenly into the turn-circle-circumference.

Use the equation: Distance = Wheel Circumference X Rotations

Distance will be the turn-circle-circumference you calculated above. You get to pick the number of rotations to make a reasonably sized wheel.

2a. Provide the filled out equation for determining the wheel circumference. (Filled out means put the numbers in the equation.)

2b. Provide the circumference of the wheel in meters. (Write the answer you calculated for 2a above.)

(How to find the answer for 3a & b) Third: prove that your wheel is a multiple of the turn-circle-circumference.

Fill out this equation by putting the number of rotations and wheel circumference into it. Then solve for distance.

Use the equation: Distance = Wheel Circumference X Rotations

3a. Fill out an equation that proves your wheel circumference is a multiple of your turn-circle.

3b. What was the result of the calculation above. Does this number equal the turn-circle-circumference in 1b above?

Make a simplified model of a center-pivot irrigation system consisting of only a wheel and axle. The path that the wheel travels is the turn-circle circumference. 1. Draw a large turn circle on 1 or 1.5 sheets of paper. 2. Make an axle the radius of the turn circle. 3. -Put a disk of paper on the end of an axle for a wheel. (Use a disk shaped rather than cylinder shaped wheel.)

4. Make a wheel with a circumference that is a multiple of the turn-circle’s circumference. 5. To test this, a mark on the wheel and a mark on the turn-circle will line up before and after the wheel rolls exactly once around the turn-circle.