A manufacturer has determined that the total cost $$ C $$ of operating a factory Is $$ s $$.inn $$ C=1.5 x^{2}+45 x+15000 \text {, } $$ where $$ x $$ is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost par unit is $$ \frac{C}{x} $$ ). $$ x=\square $$ units

Question

UpStudy AI · Accepted Answer

Define the average cost per unit as
$$
A(x) = \frac{C(x)}{x} = \frac{1.5x^2 + 45x + 15000}{x} = 1.5x + 45 + \frac{15000}{x}.
$$

To minimize $$ A(x) $$, take its derivative with respect to $$ x $$ and set it equal to zero.

1. Differentiate:
   $$
   A'(x) = \frac{d}{dx}\left(1.5x\right) + \frac{d}{dx}(45) + \frac{d}{dx}\left(\frac{15000}{x}\right) = 1.5 + 0 - \frac{15000}{x^2}.
   $$
   Thus,
   $$
   A'(x) = 1.5 - \frac{15000}{x^2}.
   $$

2. Set the derivative equal to zero:
   $$
   1.5 - \frac{15000}{x^2} = 0.
   $$

3. Solve for $$ x $$:
   $$
   1.5 = \frac{15000}{x^2} \quad \Longrightarrow \quad x^2 = \frac{15000}{1.5} = 10000.
   $$
   Taking the positive square root (since production must be positive):
   $$
   x = \sqrt{10000} = 100.
   $$

Thus, the average cost per unit is minimized when 
$$
x = 100 \text{ units}.
$$
The average cost per unit is minimized at 100 units.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions