A manufacturer has determined that the total cost of operating a factory Is .inn

where 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 ).
units

Ask by Harper Wade. in the United States

Mar 22,2025

Question

A manufacturer has determined that the total cost of operating a factory Is .inn

where 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 ).
units

Ask by Harper Wade. in the United States

Mar 22,2025

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.

A manufacturer has determined that the total cost of operating a factory Is .inn

where 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 ).
units

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

A manufacturer has determined that the total cost of operating a factory Is .inn where 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 ). units

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

A manufacturer has determined that the total cost of operating a factory Is .inn

where 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 ).
units