Math for Farmers

Most math problems posed in school textbooks are created in order to use whatever new math concept the students are learning at the time. This leads to math problems that "fit" perfectly with the step-by-step process found in the textbook but don't relate to any real-life situation. However, here we have a real math problem, from a real person, working a real job, that illustrates some basic geometry principles.

My uncle, who is a farmer, asked me to solve this problem for him:

If it takes 15 minutes to drive a tractor around the perimeter of a field, driving at 6 mph, how many acres is the field? Try to figure this out on your own. If you don't know what an acre is, you can check this Wikipedia article. The answer is after the jump.



There isn't enough information to solve the problem, unfortunately. An "acre" is a measure of area: an acre is the same as 43,560 square feet or 4,840 square yards. My uncle has given us enough information to figure out the perimter of the field, but there are several shapes with the same perimeter that have different areas. Both of the shapes below have a perimeter of 20 cm, but the square shape has more than twice the area of the skinny rectangle.

Even though we can't solve the problem we can figure out a few things. Try these on your own:

1. What is the perimeter of the field?
2. We know that shapes with the same perimeter might have different areas, but what is the greatest possible area the field could have? What shape would the field have to be in order to have the greatest possible area?


I told my uncle that if the field was an exact square, then the field would be 90 acres. (This is NOT the answer to number 2!) Here's how I figured this out:

If you've done #1, you know the perimeter is (6 mph) * ( 1/4 hour) = 1.5 miles. 1 mile is 5280 feet, so 1.5 miles is 1.5 * 5280 = 7920 feet. If we divide this by 4, we'll have the length of one side of the field. 7920 / 4 = 1980 feet. To find the area of the field, we'll take 1980*1980 = 3,920,400 square feet. (Remember the area of a square is the length of one side times itself.) This is the area of the field. To convert to acres, we'll need to divide by 43,560 square feet, which is the number of square feet in one acre. 3,920,400 / 43,560 = 90. So there we have it. If the field were a square, it would be 90 acres.

Wait a second! 90 = 15 * 6. We were given that the tractor moves at 6 mph and takes 15 minutes to drive around the perimeter, and 15*6 = 90. Is there a shortcut to this problem? If there is a shortcut, it would save time for a lot of farmers who need to estimate the acreage of their fields. See if you can figure out if this shortcut method always works.

3. Is the acreage of a square field ALWAYS (the number of minutes it takes to drive a tractor around the perimeter of the field) * (the speed of the tractor)?