applications of derivative #2

[Cherry]

Suppose the average costs of a mining operation depend on the number of machinges used, and average cost given by

(there is a line over the C) C(x) = 2900x+1,278,900/x, x>0

Where x, is the number of machines used.
a. Find the critical values for (there is a line over the C) C(x) that lie in the domain of the problem.
b. Over what interval in the domain do average cost decrease?
c. Over what interval in the domain do average cost increase?
d. How many machines give minimum average cost?
e. What are the minimum average cost?

 

[Guido]

Solution

Solution:

C=ax+b*x^(-1/2)
dC/dx=a-1/2b*x^(-3/2)=0
x^(3/2)=b/(2a)
x^3=[b/(2a)]^2
x={[b/(2a)]^2}^(1/3)
if a=2900
b=1278900
x=34.548
check if minimum
d^2C/dx^2=(-1/2)(-3/2)b*x^(-5/2)
if x is positive d^2C/dx^2 is positive
so this is a minimum on C
but x=34.548
& you can't have a fraction of a machine so compare x=34 and 35 to see which is least using
C=ax+b*x^(-1/2)