CHRISTMAS TREE FARM

Cindy Boyd Anthony Ice Donna Osborn Margaret Sanders

Class: Geometry or Algebra II

Materials: Calculator, Student Sheets, Ruler

Goals: To apply triangle relations. To apply area formulas. To make logical plans. To use Algebraic skills in making future plans.

Time

Required: Project 1: Two days. Project 2: Two days, oral presentations may take more time.

Background: Students should be familiar with triangle relationships, area, and should be able to perform Algebra I level calculations.

Setting: Your financial advisory company has been hired to solve the following problem.

Problem: Project 1: A farmer wants to grow Christmas trees on his land. He knows that the trees must be planted at least 8 feet from each other in order to grow well. For easy cultivation and care, he prefers that they be planted in parallel rows. Provide a sample planting chart.

Project 2: The farmer realizes that no trees can be sold in the first four years. However, the farmer would like some advice on how many trees he should sell in each year 5 - 9. An expense chart is provided. The first four years= expenses should be spread over year 5 - 9. Determine the number of trees to be sold each year.

Extensions: 1. Use a more complex drawing of the land. Consider drawing ponds and woods which are not polygons or circles. Also ask the students to make access drives for the farm.

2. Simulate the actual number of trees sold each year. One way to do this simulation is with a die. Each roll of the die represents the percentage of tree sold that year.

 ROLL Percentage of Trees Sold Out of the Number Projected 1 NO HELP AT CHRISTMAS TIME so 70% projected sales 2 BEAUTIFUL WHITE CHRISTMAS so 90% of projected sales are sold 3 BAD WEATHER so 40% of projected sales are sold 4 TOUGH COMPETITION so 60% of projected sales are sold 5 GOOD YEAR so 80% of projected sales are sold 6 GREAT YEAR so 100% projected sales are sold

3. Make the income spread evenly over the five years.

4. Include the cost of the land in your expense calculations.

Teacher Notes:

1. Student reports should explain the reasoning behind the number of trees they chose to sell.

2. Contact a local Christmas tree farm to find out how they plant and harvest their trees.

3. Students may have difficulty in calculating the area per tree.

4. Use spreadsheets to do the calculations for Project 2.

STUDENT SHEET

PROJECT 1

123 Country Lane

Shultzville, IN 45678

Infinite Financial Company

3.14 Accounting Lane

Mathtropolis, California 90210

Dear Sirs:

I own a farm in Southern Indiana, and I am interested in converting it to a Christmas tree farm. I have enclosed a scale drawing of my land. I need a proposal which considers the following points:

o Trees must be planted 8 feet apart

o Trees must be planted in parallel rows

o There are 43,5000 ft2 in an acre.

o Size in acres of the part of my farm which can be planted with Christmas trees

o Total number of tees to be planted

o A sample plot of how I should plant the trees

Thank you for your assistance in this endeavor.

Sincerely,

Charlie Brown

Charles Brown

Farmer

cc: Linus

Lucy

Snoopy

________________________________________________________________________________

Your creative team will submit a written report which contains

o A description of his request

o A list of assumptions

o A recommendation for Farmer Brown (including the sample plot)

o An explanation of your recommendation

STUDENT SHEET

PROJECT 1

MAP OF THE BROWN FARM

STUDENT SHEET

PROJECT 1

CLOSE-UP VIEW OF SAMPLE PLOT

In the plot above show a close-up view of 3 rows of trees. Show the distance between the rows and the distance between the trees. Briefly describe your planting arrangement. (Show all calculations used to determine the number of trees per acre.)

STUDENT SHEET

PROJECT 2

123 Country Lane

Shultzville, IN 45678

Infinite Financial Company

3.14 Accounting Lane

Mathtropolis, California 90210

Dear Sirs:

I would like for you to make an on-site visit. Please bring your nine-year plan that will estimate my income. I have included yearly charts of the current market prices for Christmas trees and the projected costs for operating the tree farm. The numbers do not include the cost of my mortgage or other expenses.

I cannot sell any trees until they grow at least four years, which means I will have no income until the fifth year. I would like to spread the initial start-up and maintenance costs evenly over the last five years. Thus I need the income of each year, 5 - 9, to at least cover my expenditures on the tree farm. At the end of the nine years I would like to have all of the trees sold.

Thank you for your continued assistance.

Sincerely,

Charlie Brown

Charles Brown

Farmer

cc: Linus

Lucy

Snoopy

________________________________________________________________________________

Your creative team will give an oral report which summarizes the written report. The written report will contain

o A description of his request

o A list of assumptions

o A recommendation for Farmer Brown

o An explanation of your recommendation

SAMPLE SOLUTION FOR PROJECT 1

Infinite Financial Company

3.14 Accounting Lane

Mathtropolis, California 90210

123 Country Lane

Shultzville, IN 45678

Dear Mr. Brown:

We are pleased that you selected Infinite Financial Company to advise you in your new endeavor. Upon receiving your request for a plan to initiate a Christmas tree farm on your land, I set our best team to work. I emphasized the importance of following your stipulations that the trees had to be planted 8 feet apart and in straight rows.

Enclosed is Infinite Financial=s proposed design for your Christmas tree farm. Included are:

a) A sample plot of three rows with the needed measurements

b) All calculations needed in finding

o The total number of acres of your farm

o The total number of acres available for trees

o The total number of trees to be planted

In our proposal, we assumed the following:

a) The houses, driveway, and creek are not included in the planting area.

b) We did not provide room for service roads.

c) All other available land on the farm is to be planted.

d) The land on which the trees will be planted is level and flat.

e) The measurements were rounded to the nearest foot.

We look forward to meeting with you in person to further develop future plans for the farm.

Sincerely,

I. B. Greedy

I. B. Greedy

President,

Infinite Financial

SAMPLE SOLUTION FOR PROJECT 1

Christmas Tree Farm Proposal

for

Charles Brown

The total number of trees that should be planted on the land is 26,418. The remainder of the proposal will explain the procedure and calculations used in arriving at our recommendation.

Part One

First, we determined the best way to plant the trees. We know that the trees must be planted in parallel rows, with trees spaced 8 feet apart. The best planting arrangement is shown below:

As you can see the trees are 8 feet apart, but the rows are approximately 6.93 feet apart. To calculate the distance between the rows, we need to know the altitude of an equilateral triangle with side 8 feet. We rounded the distance between the row to 7 feet because it is better to have the trees further apart than not far enough apart.

Part Two

The area per tree is the area of the rectangle surrounding the tree. The length of the rectangle is the same as the distance between the trees, 8 feet. The height is the same as the height of the equilateral triangle, approximately 7 feet. Therefore, the area per tree is A = |w = 8 (7) = 56 square feet.

SAMPLE SOLUTION FOR PROJECT 1

Part Three

Since there are 43,560 square feet in one acre, the number of trees per acre is 43560 ï€© 56 ï€® 777 trees. We rounded down on this operation because you can=t plant a part of a tree and there is not room for another whole tree.

Part Four

Using the given map scale, we can calculated the actual dimensions of the farm. The measurements are shown on the attached scale drawing.

The shape of the entire farm is a rectangle, so A = |w. The length is 1400 feet and the width is 1200 feet, so the area is 1,680,000 square feet.

The areas of the land which will not be planted is shown below.

Farm House and Yard: The shape of the farmhouse and yard is a rectangle, so the A = |w. The length is 350 feet and the width is 300 feet, so the area is 105,000 square feet.

Creek: The shape of the creek is a parallelogram, so the A = bh. The base is approximately 1250 feet and the height is approximately 25 feet, so the area is 31,250 square feet.

Driveway: The shape of the driveway is a rectangle. The length is 600 feet and the width is 50 feet, so the area is 30,000 square feet.

Farm House: The shape of the farmhouse is a rectangle. The length is 200 feet and the width is 150 feet, so the area is 30,000 square feet.

The total area which can not be planted is the sum of all the areas listed above, 196,250 square feet.

Part Five

The total acres which can be planted is the difference between the total number of acres of your land and the number of acres which can not be planted with trees. 1,680,000 square feet - 196,250 square feet = 1483750 square feet. This can now be changed into acres by dividing 1483750 by 43,560 ï€® 34 acres. We rounded down in this case because we do not have another complete acre to plant.

Part Six

The total number of trees can now be calculated by multiplying 34 acres by 777 trees per acre to give you 26,418 trees to be planted.

SAMPLE SOLUTION FOR PROJECT 1

6 inches = 1200 feet

1/8 in. = 25 ft.

7 in. = 1400 ft.

STUDENT SHEET

PROJECT 2

Selling Price Per Christmas Tree

 YEAR 1 - 4 5 6 7 8 9 COST PER TREE CAN=T SELL \$20 \$30 \$50 \$60 \$65

Expenses Per 100 Trees

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Planting \$100 - - - - - - - - Mowing/ Spraying \$8 \$8 \$8 \$8 \$8 \$8 - - - Trimming - - - - - \$7 \$7 \$7 - Cutting - - - - \$5 \$5 \$5 \$5 \$5 Machinery \$25 \$25 \$25 \$25 \$25 \$25 \$25 \$25 \$25 Labor \$21 \$12 \$12 \$12 \$14 \$17 \$8 \$8 \$5 Clerical \$7 \$7 \$7 \$7 \$7 \$7 \$7 \$7 \$7

STUDENT SHEET

PROJECT 2

STUDENT DATA SHEET

Following your proposal, Farmer Brown should plant ___________ trees.

The fixed cost for maintaining 100 trees for the first 4 years is ________.

The fixed for maintaining all of your trees for the first 4 years is ________.

Spreading the cost over 5 - 9, the fixed expense per year is ________________.

 Fixed Expense Number of Trees Expense per 100 trees Expense for the year Total Expense Trees Sold for Year Selling Price for year Gross Profit Net Profit 5 6 7 8 9

TOTAL PROFIT ___________

SAMPLE SOLUTION FOR PROJECT 2

Christmas Tree Farm Proposal

for

Charles Brown

The projected total income for the Christmas tree farm after nine years is \$1,214,025. The chart below shows the total number of trees sold each year to reach the projected income.

 Year Trees Sold for Year Net Profit 5 2614 \$ 19,944 6 3922 \$ 84,486 7 5229 \$234,362 8 6537 \$367,851 9 8116 \$507,382

The following assumptions were made figuring these calculations.

1. All money calculations are rounded to the nearest dollar.

2. All trees grow at the same rate.

3. There is a market for the number of trees that you want to sell.

4. No tree dies.

5. The expenses and selling prices remain consistent with the original projected values.

Since the tree farm is going to be your only source of income, it is important that you have a substantial income each of the five years. You will note that we have allowed for a growing profit over the given time period. Any excess income can be easily invested in certificates of deposit, savings accounts, stocks, and bonds to increase your income. We will be happy to assist you in choosing the appropriate investments to build your revenues.

On the next page is a spreadsheet with an explanation of the calculations for the expenses and the profits.

SAMPLE SOLUTION FOR PROJECT 2

SAMPLE STUDENT DATA SHEET

Following your proposal, Farmer Brown should plant 26,418 trees.

The fixed cost for maintaining 100 trees for the first 4 years is \$317.

The fixed for maintaining all of your trees for the first 4 years is \$83,745.

Spreading the cost over years 5 - 9, the fixed expense per year is \$16,749.

 Fixed Expense Number of Trees Expense Per 100 Trees Expense For The Year Total Expense Trees Sold For Year Selling Price For year Gross Profit Net Profit 5 \$16,749 26,418 \$59 \$15,587 \$32,336 2614 \$20 \$ 52,280 \$ 19,944 6 \$16,749 23,804 \$69 \$16,425 \$33,174 3922 \$30 \$117,660 \$ 84,486 7 \$16,749 19,882 \$52 \$10,339 \$27,088 5229 \$50 \$261,450 \$234,362 8 \$16,749 14,653 \$52 \$ 7,620 \$24,369 6537 \$60 \$392,220 \$367,851 9 \$16,749 8,116 \$42 \$ 3,409 \$20,158 8116 \$65 \$527,540 \$507,382

TOTAL PROFIT \$1,214,025

o The expense per 100 trees is calculated by totaling the individual expenses for that year.

o The expense for the year is calculated by divididing the number of trees by 100 and then multiplying by the expense per 100 trees.

o The total expense is calculated by adding the fixed expense and the expenses for the year.

o The gross profit is calculated by multiplying the number of trees sold that year by the selling price for that year.

o The net profit is calculated by subtracting the total expense from the gross profit.

Funded in part by the National Science Foundation and Indiana University 1995