Respuesta :
Answer:
The production of wheel per day is 74 which gives lowest average cost per wheel.
The minimum average cost is $168.72.
Step-by-step explanation:
Given function of average cost is
[tex]C(x)= 0.03x^3-4.5x^2+171x[/tex]
Differentiating with respect to x
C'(x)= 0.09 x² -9.0 x+171
Again differentiating with respect to x
C''(x) = 0.18 x -9.0
To find the minimum average cost, first we have to set C'=0.
The function's slope is zero at x=a, and the second derivative at x=a is
- less than 0, it is critical maximum.
- greater than 0, it is critical minimum.
Now ,
C'=0
⇒ 0.09 x² -9.0 x+171=0
⇒x = 74.49, 25.50
[tex]C''(x)|_{x=74.49} = 0.18 (74.49)-9.0=4.41>0[/tex]
[tex]C''(x)|_{x=25.50} = 0.18 (25.50)-9.0=-4.41<0[/tex]
Therefore at x= 74.49≈ 74, the average cost is minimum.
The production of wheel per day is 74 which gives lowest average cost per wheel.
The minimum average cost [tex]C(x)= 0.03x^3-4.5x^2+171x[/tex]
[tex]=(0.03 \times 74^3)-(4.5 \times 74^2)+(171\times 74)[/tex]
=168.72
[Assume the average cost is in dollar]
The minimum average cost is $168.72.
