Eat a Peach!

Property Description: From a starting point north, the east side of your property goes due south 960 feet. From the southernmost tip of this line your southern property border extends from east to west 1440 feet. From the western end of the southern boundary, your property line extends due north for 1200 feet before it makes a 45-degree turn to the northeast. The line that extends from the point at which the northeast turn begins to the northernmost point of your property is called the “Georgia Fence Line.” If you were to travel due east from the north end of your south-to-north, west boundary line, you could travel 1200 feet before you stepped off of your property. The route you traveled would form the southern base of an isosceles right triangle. The height of this triangle will take you to the northern boundary of your property, and the line extending southwest from that point and connecting to the south-to-north line composing your western boundary will compose your northwest boundary. The northeast side of your property extends southeast from the northern boundary of your property to the northern end of the north-to-south line that defines the eastern edge of your property and is called the “Texas Fence Line.”

Your Tools: One piece of string, markers, dry erase boards, graph paper

Your Preparation: 1. Re-write this property description to make it easier to understand. 2. Draw a map of your property to scale on the graph paper. Include in your drawing the measurements of each side of your property. 3. Brainstorm and illustrate the numerous geometric and numerical relationships you can determine based on the information given in the property description. The more relationships you find, the better off you will be. 4. Determine the area of your property in square feet, square yards (3 feet = 1 yard), acres (1 acre = 43,560 square feet), and square miles (1 square mile = 640 acres). 5. Determine the perimeter of your property. 6. Determine the shortest distance from the southwest corner of your property to the Texas Fence Line, the shortest distance from the southeast corner of your property to the Georgia Fence Line, and the shortest distance from the northern tip of your property to the southern property line. Problems You Need to Solve: 1. Watering Before planting anything on your property, you feel that you need to soften the ground – and rain is nowhere in sight. The only type of irrigation system you have is a single-arm, center-pivot irrigation system. At most, you can only water from one pivot point for one rotation of the system. You do not want to water the property of others and you certainly do not want to run over a fence.  To water the greatest amount of your property and assuming the system does not water outside the length of the arm, where would you place the center pivot, and how long would that arm be?  What percentage of the field would not be watered?  If you were allowed two pivot points, but still only one complete rotation of the system, how much of the field would remain dry? How much of the field would be watered twice?  What percentage of the allowed rotation would be used on the first pivot? The second pivot? Would remain unused?

2. Planting After you have softened the ground and terraced your property, you are ready to plant your peach orchard. Due to water run-off you have decided it is best to plant your trees in diagonal rows from northwest to southeast. Your first tree will be planted on the southwest corner of your property. From that point, you measure 10 yards north on the western border and 10 yards east on the southern border. The diagonal formed by connecting these two points will define the line on which you plant your second row of trees. On this row you will plant two trees, one on the western boundary and one on the southern boundary. From the point at which these trees are planted you will measure 10 more yards north on the western border and 10 more yards east on the southern border. Connecting these points will form the line on which you will plant your third row of trees. On this row you will plant three trees, one on each of the boundary lines and one at the mid-point. You will continue this pattern of parallel lines and “checkerboard” planting all the way to the Texas Fence Line.  Making the assumption that you will not plant directly on the Texas Fence Line, how many rows of trees will you have upon completion?  How many trees will you have planted?  On which row will you plant the most trees, and how long is that row?  What is the length of and how many trees will be planted on every tenth row?  On a line that bisects the southwest corner of your orchard and is perpendicular to the Texas Fence Line, how far apart will your rows be?  What is the length of the line in the question above, and how many trees will be planted on that line?

3. Purchasing Peach Trees You know that quality peach saplings cost $1.50 for each of the first one hundred with a 10% discount for the next hundred, 10% off that price for the next hundred, and so forth.  How much would you pay for each of your last one hundred peach saplings?  What would be the average cost of each of your peach trees?  Create a formula for calculating the above information.

4. Peach Farming Problems of Economics Peach farming is not without its problems, and you will want to make your peach orchard profitable in the end. You know that two trees in every hundred will die every year. Three in every hundred that live will fail to produce any peaches. Your trees won’t produce any peaches the first year. During the second year your trees will produce half the peaches they are capable of producing when they are mature, 75% the third year, and will reach full production levels the fourth year. A mature tree produces an average of 50 pounds of fruit per year, but ten percent of that will be substandard. You will get at least fifteen cents per pound for standard peaches at market, but not more than twenty cents per pound. Your trees will live for 10 seasons. Peach production will fall to 75% the eighth year, 50% the ninth year, and you will produce no fruit the tenth year.  Create a formula for calculating the gross income for standard peaches for each year.  Produce a graph that would show peach production and gross income over a ten year period.