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Keeping a Cow Happy

Note: Do each of these problems in order...they get increasingly complex and require some creative thinking.

Problem 1: A cow is tied to a stake by a rope 15-feet long. Over how much ground can she graze?

Problem 2: In the previous problem, suppose we want the same cow to be able to graze over 1000 sq. ft. How long must the rope be?

Problem 3: Now the cow is tethered at the end of a 50-foot rope, which is fastened to a corner of a rectangukar barn. If the barn is 25 feet by 60 feet, over how much ground (i.e. area) may the cow graze?

Problem 4: In the previous problem, suppose we want the same cow to be able to graze over 1000 sq. ft. How long must the rope be?

Problem 5: In Problem #3, suppose that the rope was 85 feet long. How much ground could the cow graze?

Problem 6: In Problem #3, suppose the barn was a parallelogram, with the cow anchored at the corner having a 45o angle? Also, would the cow graze more/less/same amount if anchored at the 135o angle>?

Problem 7: In Problem #3, suppose the barn was circular with a radius of 25 feet. How much ground could the cow graze?

Problem 8: Make up your own creative variation for this hungry cow!

Source: Adapted from J. Thompson's Geometry for the Practical Man, 1934, pp. 290-291

Hint: Draw pictures...if necessary, break the problems into sub-problems (or calculations of sub-areas).

Solution Commentary: The first answer is about 707 sq. ft.

The second answer is almost 18 feet long.

In the third problem, do you see how the desired area is basically 3/4 of a circle of one radius and 1/4 of a circle of a lesser radius?...now proceed on your own.