1. The problem statement, all variables and given/known data Chemical-plant equipment costs rarely vary in proportion to (i.e., linearly with) size. In the simplest case, cost varies with size according to the allometric equation, C = aSb The exponent is typically between 0 and 1. For a wide variety of equipment types, it is approximately 0.6. (a) For 0 < b < 1, show that cost per unit size decreases with increasing size, resulting in an economy of scale. (b) Consider the case of a spherical storage tank. The size is commonly measured by internal volume Vt. Show that b = 2/3. On what parameters or properties would you expect the quantity a to depend? 2. Relevant equations formulas for derivative 3. The attempt at a solution For part a, I calculated the derivative of C/S with respect to the size, which equals to (b-1)*a*Sb-2. As the value of b-1 is negative, the value of this derivative is always negative, and so the increase in size would result in a decrease in the cost per unit size. I don't understand what part b is asking about though. Compared with part a, it doesn't give anything apart from telling me that the tank is spherical. I have completely no clue about how to obtain a specific value for b without any additional information. Any help would be much appreciated!