1. The problem statement, all variables and given/known data X,Y,Z(coordinates) What function corresponds to the best tunnel shape? g = 9.8 m/s^2(earth gravity) 2. Relevant equations F(x)=Y G(x)=X^2 in the xy plane G(z)= sin(X) in the xz plane H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane) 3. The attempt at a solution I have thought about the shape of a tunnel and as far as I understand, the less the change of Y, the better since if Y changes, U changes(i am using U for upward force), namely as Y gets more negative, U gets more positive. I think I need calculus to solve this and maybe none of those functions are the best. However, the further negative Y gets, the more the velocity increases and the more positive Y gets, the more velocity decreases. From this point of view the parabolic sinusoid is best. And the higher the velocity the higher U is. Is there a way I can solve this problem without calculus? If so how? X and Z don't really play a part into the upward force that is required for a tunnel not to collapse from gravity. I can probably find the average density of soil to determine how much mass of soil there is above the tunnel and thus the maximum downward force it can withstand with nothing but the soil itself providing upward force and thus determine how much more upward force is needed at some depth Y.