The Nearest Faraway Place Rally 2017

Route Card 11 (Easy Plot)

Route Card 11 (Easy Plot)
Display Map Solution     

Plotting Boundaries: (E 551000, N 261400) to (E 564500, N 252000)
This is a timed to the Second Section.

TC1 @ 560819 255452
TC2 @ ?

The "dummy" route runs from TC1 anti-clockwise on a loop back to TC1.

Four sections of road 1) TC1 to (562694 254887), 2) to (563044 255798), 3) to (561348 256767), 4) to TC1. The markers are exactly in the centre of each junction.

Consider this route to be flat, i.e. there are no gradients. You can get the "flat" distances between junctions from the Plot Route screen by marking a dummy route.

However, I've been a bit sneaky. I've erected "up" ramps on sections 1) and 2), and "down" ramps on sections 3) and 4). The ramps provide a uniform ascent or descent on each section.
1) Gradient up is at 38.6 degrees.
2) Gradient up is at 41.7 degrees
3) Gradient down is at 35.9 degrees.
4) Gradient down is at the appropriate number of degrees to return you to TC1 at the same level as you started.

Using the ramps, how far do you travel from TC1 to TC1?

Erase your dummy route and mark a single point for TC2 at the distance above on the yellow road that heads WNW from TC1 crossing the A1304 to Dunsley Corner. Submit your route.

Route Card Commentary

You could have used: http://www.analyzemath.com/Geometry_calculators/right_triangle_calculator.html

Here's the maths:

1) Flat distance 1.272 @ 38.6 degrees gives an ascent of 1.01542425 and a ramp distance of 1.62759651.

2) Flat distance 0.612 @ 41.7 degrees gives an ascent of 0.54527210 and a ramp distance of 0.81967412.

Total ascent = 1.56069635

3) Flat distance 1.249 @ 35.9 degrees gives a descent of 0.90412521 and a ramp distance of 1.54189604.

4) Descent required to TC1 is 1.56069635 - 0.90412521 = 0.65657114. Flat distance 0.887 and descent 0.65657114 gives a ramp distance of 1.10356452.

Total distance for TC2: 1.62759651 + 0.81967412 + 1.54189604 + 1.10356452 = 5.09273119

Here's a Javascript program which gives the asnswer.
var f1 = 1.272, f2 = 0.612, f3 = 1.249, f4 = 0.887;
var d1 = 38.6, d2 = 41.7, d3 = 35.9, d4;
var a1 = f1*Math.tan(d1 * Math.PI/180);
document.write("Ascent 1 = "+a1);
var l1 = f1/Math.cos(d1 * Math.PI/180);
document.write("Distance 1 = "+l1);
var a2 = f2*Math.tan(d2 * Math.PI/180);
document.write("Ascent 2 = "+a2);
var l2 = f2/Math.cos(d2 * Math.PI/180);
document.write("Distance 2 = "+l2);
var ta = a1 + a2;
document.write("Total Ascent = "+ta);
var a3 = f3*Math.tan(d3 * Math.PI/180);
document.write("Descent 1 = "+a3);
var l3 = f3/Math.cos(d3 * Math.PI/180);
document.write("Distance 3 = "+l3);
var a4 = a1 + a2 - a3;
document.write("Descent Required = "+a4);
var l4 = Math.sqrt(Math.pow(a4,2)+Math.pow(f4,2));
document.write("Distance 4 = "+l4);
var tl = l1 + l2 + l3 + l4;
document.write("Total Distance = "+tl);