Homework Help: Chemical-plant equipment costs vary to size

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1. Jan 10, 2016

Ortanul

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.

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 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!

2. Jan 10, 2016

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?

3. Jan 10, 2016

Ortanul

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

4. Jan 10, 2016

Ortanul

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

5. Jan 10, 2016

Staff: Mentor

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