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 15feet 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 50foot 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 45^{o} angle? Also, would the cow graze more/less/same amount if anchored at the 135^{o} 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. 290291
Hint: Draw pictures...if necessary, break the problems into subproblems (or calculations of subareas).
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.
