| 3. (a) Solve (1/2)d + (1/4)d = 1, taking x = (1/2)d, obtaining
x + x2 = 1. The solution of the quadratic equation is x = (-1 + √5)/2, so
d = Log[(-1 + √5)/2)/Log(1/2). |
| (b) Solve (1/2)d + 2⋅(1/4)d = 1, taking x = (1/2)d, obtaining
x + 2x2 = 1. The solution of the quadratic equation is x = 1/2, so
d = Log(1/2)/Log(1/2)= 1. |
| (c) Solve (1/2)d + N⋅(1/4)d = 1, taking x = (1/2)d, obtaining
x + N⋅x2 = 1. The solution of the quadratic equation is x = (-1 + √(1 + 4n))/(2n), so
d = Log((-1 + √(1 + 4⋅N))/(2N))/Log(1/2) |
| (d) The largest dimension of a subset of the plane is d = 2. So we solve
2 = Log((-1 + √(1 + 4⋅N))/(2N))/Log(1/2) for N: |
| 2 Log(1/2) | = Log((-1 + √(1 + 4⋅N))/(2N)) |
| Log(1/4) | = Log((-1 + √(1 + 4⋅N))/(2N)) |
| 1/4 | = (-1 + √(1 + 4⋅N))/(2N) |
| N | = -2 + 2*√(1 + 4⋅N)) |
| N + 2 | = √(4⋅(1 + 4⋅N)) |
| (N + 2)2 | = 4⋅(1 + 4⋅N) |
| N2 - 12⋅N | = 0 |
|
| So N = 12. |