In #29 you were given the length of a spiral. In #30 you were given the area of a spiral. For this cache you will be given both the length and the area so it should be easier. Right?
Probably the most commonly occurring spiral is the spiral of Archimedes. You can observe it in coiled ropes, the grooves of records, and of course in crop spirals, such as the one due to Archie of Medes described below. In fact, the spiral appears well before recorded history, for example in the ancient temples of Ireland and Malta. Around 325 BC the great mathematician Archimedes was able to obtain significant results regarding its area, tangents and curvature; hence it bears his name.
Spirals are often expressed in polar coordinates in which a point in the plane is given by the pair (r, θ) where r is the distance from the origin to the point and θ the angle measured counterclockwise from the positive x-axis.
In the simplest form of the spiral of Archimedes, the two quantities r and θ are equal , i.e. the equation is just
r = θ. Thus at the angle π/2 the distance to the point is also π/2. At θ = π, we have r = π, and so on.
To generalize slightly let's scale everything up by a factor of M, so that lengths will be multiplied by M and the areas by M2. Then we have the following equations for the distance, area and arclength, resp.:
r = M θ
A = (1/6) M2θ 3
L = (M/2) [θ √ (1+θ2) + ln (θ +√ (1+θ2))]
in which ln is the natural log function. These equations are much messier if one were to use the common log (base 10) or angles measured in degrees rather than radians. On the other hand, the formula for L simplifies if one uses sinh-1, the inverse hyperbolic sine function.
So, from these equations, all you really need to know to locate the cache are the area and arclength which are
A = 135.5 hectares and L = 3519 meters..
Hints: To get started, here are the first digits of R, M and θ: R=1xxx, M=2xx.x, θ=4.xxx. As indicated here, four digits of accuracy should suffice.
Some pictures may help. The figure on the left above shows the basic polar coordinates, the distance r and the angle θ (theta). Notice that θ is measured counterclockwise from the positive x-axis (east). Let ψ (psi) denote the angle usually used in geocaching which is measured clockwise from the north. These two angles are complementary, that is their sum is π/2 (90°), possibly plus or minus some integer multiple of 2π (360°)
The figure on the right above shows the first loop of the basic spiral r = θ, which at 2π reaches the circle of radius 2π. Use the formula with M=1 to check that the area of this first loop is one third the area of the circle. Although conjectured a century earlier this was first proved by Archimedes in 325 B.C.
If one keeps on plotting the spiral for larger values of θ, one gets the familiar looking plot shown on the left below. By the way, the two spirals considered in the previous two geocaches in this series approach an Archimedian spiral as they become larger and larger.
The figure on the right below depicts the area and length of a typical spiral similar to the one under consideration in this puzzle cache. Note that that length is just for the curved part of the spiral and so does not enclose the area. The so-called full spiral is listed with the images to the left below, but is not shown otherwise..
Finally consider the picture centered at the end of the descriptions. It shows the crop spiral of Archie of Medes. The spiral of Archimedes can an be simply constructed. Moving counterclockwise from the origin as the angle is incremented successively by a constant amount the distance back to the origin is incremented by a proportionate amount. Archie overlapped two such spirals. It's said that that attracts parrots
Keywords: Spiral of Archimedes, polar coordinates, area of the spiral of Archimedes, arclength of the spiral of Archimedes.
Verification: After rounding, when the solution to the puzzle is offset or projected from the given coordinates to obtain the location of the cache the VK easting and northing will end in 0 and 5, resp . (Similarly the lat/lon coordinates for that matter will end in 5 and 0.) Thus your solution to be entered into the geochecker is of the form VKabcd0fghi5. If successful you will be given the lat/lon coordinates as well as a hint to the hide. Please note that to reach the cache you do not need to cross any fields or fences, not even any ditches. You will only trample a few weeds.
You can validate your puzzle solution with certitude.
The first ten solvers are
1. pfalstad Tue, 11 Jul 2017 0:30:53
2. rickrich Tue, 11 Jul 2017 7:07:59
3. foundinthewild Tue, 11 Jul 2017 16:57:12
4. ctc128 Tue, 18 Jul 2017 21:06:56
5. PackADad Fri, 15 Sep 2017 19:46:59
6. rohrerhj Wed, 11 Oct 2017 15:25:17
7. salsman Thu, 2 Nov 2017 19:52:22
8. pcc322 Sat, 11 Nov 2017 18:57:26
9. PA-20 Tue, 28 Nov 2017 15:43:40
10. fish2007 Mon, 15 Jan 2018 19:35:07