Answer to Question #275555 in Calculus for sumu

Question

A manufacturing process costs RM 6500 to set up for one year’s use. If items cost RM 85 each to produce and other costs amount to 3.5 x2, where x is the production in hundreds, find the level of production that will minimize the cost per item over the year. What will the total cost amount to at this level of production?

Answer 1

$$C(x)=6500+85x+3.5(0.01x)^2, x>0$$

$$C(x)/x=6500/x+85+3.5(0.01)^2x$$

$$(C(x)/x)'=-6500/x^2+0.00035$$

Find the critical number(s)

$$(C(x)/x)'=0=>-6500/x^2+0.00035=0$$

$$x^2=18571428.5714$$

$$x=\pm4309.458$$

We consider $x>0.$ The function $C(x)/x$ has the absolute minimum for $x>0$ at $x=4309.458$

$$C(8062)=6500+85(8062)+$$

$$C(4309)/4309=6500/4309+85+0.00035(4309)$$

$$\approx88.01662$$

$$C(4310)/4310=6500/4310+85+0.00035(4310)$$

$$\approx88.01662$$

$$C(4309)=6500+85(4309)+0.00035(4309)^2$$

$$\approx379263.62$$

$$C(4310)=6500+85(4310)+0.00035(4310)^2$$

$$\approx379351.64$$

The level production of $4309$ or $4310$ items will minimize the  cost per item over the year.

The  total cost amount is RM $379263.62$ at level production of $4309.$

The  total cost amount is RM $379351.64$ at level production of $4310.$