Question

the total cost, in dollars, to order x units of a certain product is modeled by ( c(x)=7x^{2}+252 ). according to the model, for what size order is the cost per unit a minimum?
a an order of 1 unit has a minimum cost per unit.
b an order of 6 units has a minimum cost per unit.
c an order of 84 units has a minimum cost per unit.
d an order of 252 units has a minimum cost per unit.
answer b

Explanation:

Step 1: Define Cost per Unit

The cost per unit (average cost) \( \bar{C}(x) \) is the total cost \( C(x) \) divided by the number of units \( x \). So, \( \bar{C}(x)=\frac{C(x)}{x} \). Given \( C(x) = 7x^{2}+252 \), we have \( \bar{C}(x)=\frac{7x^{2}+252}{x}=7x+\frac{252}{x} \) (for \( x>0 \)).

Step 2: Find Minimum of Average Cost

To find the minimum of \( \bar{C}(x) \), we can use the AM - GM inequality or calculus. Using AM - GM: for positive numbers \( a \) and \( b \), \( \frac{a + b}{2}\geq\sqrt{ab} \), with equality when \( a = b \). Let \( a = 7x \) and \( b=\frac{252}{x} \). Then \( \bar{C}(x)=7x+\frac{252}{x}\geq2\sqrt{7x\times\frac{252}{x}} \). Simplify the square root: \( 7x\times\frac{252}{x}=7\times252 = 1764 \), so \( \sqrt{1764}=42 \), and \( 2\times42 = 84 \). Equality holds when \( 7x=\frac{252}{x} \). Solve \( 7x=\frac{252}{x} \): multiply both sides by \( x \) ( \( x
eq0 \)) to get \( 7x^{2}=252 \), then \( x^{2}=\frac{252}{7}=36 \), so \( x = 6 \) (since \( x>0 \)).

Answer:

B. An order of 6 units has a minimum cost per unit.