Question

the total cost, in dollars, to order $x$ units of a certain product is modeled by $c(x) = 5x^2 + 320$. 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 8 units has a minimum cost per unit.
c an order of 80 units has a minimum cost per unit.
d an order of 320 units has a minimum cost per unit.

Explanation:

Step1: Define cost per unit

Let \( C(x) = 5x^2 + 320 \) be total cost. Cost per unit \( c(x)=\frac{C(x)}{x}=\frac{5x^2 + 320}{x}=5x+\frac{320}{x} \).

Step2: Find derivative of \( c(x) \)

Differentiate \( c(x) \) with respect to \( x \): \( c^\prime(x)=5-\frac{320}{x^2} \).

Step3: Set derivative to zero

To find minima, set \( c^\prime(x) = 0 \):
\( 5-\frac{320}{x^2}=0 \)
\( 5=\frac{320}{x^2} \)
\( x^2=\frac{320}{5}=64 \)
\( x = 8 \) (since \( x>0 \) for units).

Step4: Verify minimum (optional, but for confirmation)

Second derivative \( c^{\prime\prime}(x)=\frac{640}{x^3} \), at \( x = 8 \), \( c^{\prime\prime}(8)>0 \), so it's a minimum.

Answer:

B. An order of 8 units has a minimum cost per unit.