Chemical-plant equipment costs vary to size

[Ortanul]

Homework Statement

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 = aS^b

The exponent is typically between 0 and 1. For a wide variety of equipment types, it is approximately 0.6.

1. (a) For 0 < b < 1, show that cost per unit size decreases with increasing size, resulting in an economy of scale.
2. (b) Consider the case of a spherical storage tank. The size is commonly measured by internal volume V_t. Show that b = 2/3. On what parameters or properties would you expect the quantity a to depend?

Homework Equations

formulas for derivative

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*S^b-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!

 

[256bits]

@Ortanul, Hi

What is the volume of a sphere of radius r?
Then, what other equation of a sphere could one apply that might indicate its cost with regard to its cost of material. How would you find that?

 

[Ortanul]

@265bits
Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?

 

[Ortanul]

@Ortanul, Hi

What is the volume of a sphere of radius r?
Then, what other equation of a sphere could one apply that might indicate its cost with regard to its cost of material. How would you find that?

Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?

 

[Chestermiller]

Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?

What is the volume of the shell. What do you get if you multiply this by the density of the shell metal?