»  Solutions to puzzles in my National Review Online Diary

  February 2003

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In my February diary I posed the following brain-teaser:

A lighthouse has a circular floor plan, 20 feet in diameter. On the outside wall of the lighthouse is fixed a hook. A leash is tied to the hook, with a dog at the other end. The leash is 10π feet (that is to say, a tad more than 31.4159265358 9793238462 6433832795 0288419716 9399375105 8209749445 9230781640 6286208998 6280348253 4211706798 2148086513 2823066470 9384460955 0582231725 3594081284 8111745028 4102701938 5211055596 4462294895 4930381964 4288109756 6593344612 8475648233 7867831652 7120190914 5648566923 4603486104 5432664821 3393607260 2491412737 2458700660 6315588174 8815209209 6282925409 1715364367 8925903600 1133053054 8820466521 3841469519 4151160943 3057270365 7595919530 9218611738 1932611793 1051185480 7446237996 2749567351 8857527248 9122793818 3011949129 8336733624 4065664308 6021394946 3952247371 9070217986 0943702770 5392171762 9317675238 4674818467 6694051320 0056812714 5263560827 7857713427 5778960917 3637178721 4684409012 2495343014 6549585371 0507922796 8925892354 2019956112 1290219608 6403441815 9813629774 7713099605 1870721134 9999998372 9780499510 5973173281 6096318595 0244594553 4690830264 2522308253 3446850352 6193118817 1010003137 8387528865 8753320838 1420617177 6691473035 9825349042 8755468731 1595628638 8235378759 3751957781 8577805321 7122680661 3001927876 6111959092 1642019893 8095257201 0654858632 7886593615 3381827968 2303019520 3530185296 8995773622 5994138912 4972177528 3479131515 5748572424 5415069595 0829533116 8617278558 8907509838 1754637464 9393192550 6040092770 1671139009 8488240128 5836160356 3707660104 7101819429 5559619894 6767837449 4482553797 7472684710 4047534646 2080466842 5906949129 3313677028 9891521047 5216205696 6024058038 1501935112 5338243003 5587640247 4964732639 1419927260 4269922796 7823547816 3600934172 1641219924 5863150302 8618297455 5706749838 5054945885 8692699569 0927210797 5093029553 2116534498 7202755960 2364806654 9911988183 4797753566 3698074265 4252786255 1818417574 6728909777 7279380008 1647060016 1452491921 7321721477 2350141441 9735685481 6136115735 2552133475 7418494684 3852332390 7394143334 5477624168 6251898356 9485562099 2192221842 7255025425 6887671790 4946016534 6680498862 7232791786 0857843838 2796797668 1454100953 8837863609 5068006422 5125205117 3929848960 8412848862 6945604241 9652850222 1066118630 6744278622 0391949450 4712371378 6960956364 3719172874 6776465757 3962413890 8658326459 9581339047 8027590099 4657640789 5126946839 8352595709 8258226205 2248940772 6719478268 4826014769 9090264013 6394437455 3050682034 9625245174 9399651431 4298091906 5925093722 1696461515 7098583874 1059788595 9772975498 9301617539 2846813826 8683868942 7741559918 5592524595 3959431049 9725246808 4598727364 4695848653 8367362226 2609912460 8051243884 3904512441 3654976278 0797715691 4359977001 2961608944 1694868555 8484063534 2207222582 8488648158 4560285060 1684273945 2267467678 8952521385 2254995466 6727823986 4565961163 5488623057 7456498035 5936345681 7432411251 feet) long. Assuming the dog cannot enter the lighthouse, and the surface he can travel is flat and unobstructed, what area can he cover?

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Solution

This one needs a diagram. The inner circle represents the lighthouse. The hook is at the right on the lighthouse wall. At full stretch, the dog can be 10π feet away from the hook; that's the right-most point of the diagram.

Dog on leash

If the dog heads "north" from this point (assuming, with no loss of generality, that the diagram is oriented in the usual north-south-east-west way), and still keeps the leash taut, he can walk out a quarter-circle, ending at the top of the diagram.

If he keeps on walking — he'd be heading directly due west at first — the leash starts to wrap round the wall of the lighthouse. So the radius of his path is no longer constant. It's lessening all the time, as more and more leash wraps round the lighthouse wall … until the dog himself crashes into the wall. Since the leash is 10π feet long, and that is precisely half the circumference of the lighthouse (circumference of a circle is π times its diameter, remember), the dog crashes into the lighthouse wall at precisely the west-most point of the wall, with all his leash wrapped round the north half of the lighthouse.

The same applies in the south half of the diagram, of course. We can just solve the problem for the north half, then double the answer.

So what is the area the dog can cover in the north half? Well, it's that quarter of a circle, plus that non-circular bit the dog traces out when his leash is wrapping round the wall.

The quarter circle is easy. Area of a full circle is π times square of radius, in this case π times square of 10π. That makes 100π3. We only need a quarter of that: 25π3.

To get the non-circle area, I believe you have to use calculus. (Though if there is a non-calculus solution, I'd be glad to hear it.) Consider the situation, leash at full stretch, when the leash has already wrapped round an arc of the wall that makes an angle t at the center. The angle t is of course measured in radians. Length of this arc is 10t, by elementary geometry (length of an arc of a circle). The remainder of the leash — the part that's straight — is therefore 10π − 10t, or 10(π − t).

Check:  When the angle  −  is zero, the straight part is 10π. When the angle  −  is π radians (that is, 180 degrees), the straight length is zero. Yep.

Now increase  −  by a tiny amount dt. The tangent advances correspondingly; and the angle between the advanced tangent and the original one, again by elementary geometry, is dt. The area of the shaded triangle between the two tangents, to a first-order approximation, is 50(π − t)2dt, again by elementary geometry (area of a sector of a circle). If you integrate that from t equals zero to t equals π, you get 50π3/3.

The top half of the required area is therefore 25π3 plus 50π3/3. This is 125π3/3.

The entire area is twice this (north plus south):  250π3/3.

Solution:  250π3/3, or about 2583.856390 0249850146 2302625559 1782933519 square feet.