Question 148177
If I just know Ahmed's half, the treasure has to be at the outer
edge of a circle with center at Castle Rock and radius = {{{2x + 6}}}
paces.
Vanessa's half says "walk x paces to the north, and then walk 2x + 4 paces to the east." But her instructions must put her on the edge of Ahmed's circle with
radius = {{{2x + 6}}}
The radius and the North and South walks form a right triangle, so
{{{(2x + 6)^2 = x^2 + (2x + 4)^2}}}
{{{4x^2 + 24x + 36 = x^2 + 4x^2 + 16x + 16}}}
subtract {{{4x^2}}} from both sides
{{{24x + 36 = x^2 + 16x + 16}}}
subtract {{{24x}}} from both sides
{{{36 = x^2 - 8x + 16}}}
subtract {{{36}}} from both sides
{{{x^2 - 8x - 20 = 0}}}
now solve for {{{x}}}
{{{(x - 10)(x + 2) = 0}}}
{{{x = 10}}} answer
{{{x = -2}}} reject this one (can't walk minus paces)
Now plug this into data given to check:
Ahmed's paces:
{{{2x + 6}}}
{{{2*10 + 6 = 26}}}
Vanessa's paces:
East: {{{2x + 4}}}
{{{2*10 + 4 = 24}}}
North: {{{x = 10}}}
Now does right triangle check out?
{{{(2x + 6)^2 = x^2 + (2x + 4)^2}}}
{{{26^2 = 24^2 + 10^2}}}
{{{676 = 576 + 100}}}
{{{676 = 676}}}
OK