Alan

This is modular arithmetic (clock arithmetic).  

d = (1+1288*n)/3  where n is a positive integer (not all positive integers!).

In fact the smallest positive integer for n that gives a positive integer for d is n=2, for which d = 859.

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{859}} = {\mathtt{2\,577}}$$

$${\mathtt{2\,577}} \,{mod}\, {\mathtt{1\,288}} = {\mathtt{1}}$$

Generally, n = 2+3k, where k is 0, 1, 2, ... etc. will result in a value of d that satisfies mod(3d,1288)=1

Have you written them down correctly?  These two functions are clearly not equivalent.  For example:

Lengths scale as the square root of areas.  So, as heureka has calculated,

$${\mathtt{radiusofRhea}} = {\sqrt{{\frac{{\mathtt{34}}}{{\mathtt{3}}}}}}{\mathtt{\,\times\,}}{\mathtt{1\,530}} \Rightarrow {\mathtt{radiusofRhea}} = {\mathtt{5\,150.747\: \!518\: \!564\: \!660\: \!29}}$$

Note that this is 5150km to the nearest ten kilometres.

One to one means that if 'a' goes to 'b' then 'b' goes back to 'a'.  So f^-1(12) = 5

Looks like his maximum profit is $425 a day

You don't say what x is!  I've assumed it's the height of the vertical parts.

Ok, here's another answer (different from the previous one):

L=3  A=6  S=4  E=2  R=9  H=5

LASER + REAL

$${\mathtt{36\,429}}{\mathtt{\,\small\textbf+\,}}{\mathtt{9\,263}} = {\mathtt{45\,692}}$$

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45692