# A lighting fixture manufacturer has daily production costs of C=0.25n^2- 10n+800, where C is the total daily cost in dollars and n is the number of light fixtures produced. How many fixtures should be produced to yield a minimum cost?

Oct 29, 2016

The minimum cost would be the $C$ coordinate of the vertex. Hence, we will have to complete the square.

$C = 0.25 {n}^{2} - 10 n + 800$

$C = 0.25 \left({n}^{2} - 40 n\right) + 800$

$C = 0.25 \left({n}^{2} - 40 n + 400 - 400\right) + 800$

$C = 0.25 \left({n}^{2} - 40 n + 400\right) - 100 + 800$

$C = 0.25 {\left(n - 20\right)}^{2} + 700$

The vertex is $\left(20 , 700\right)$, in the form $\left(n , C\right)$, so the minimum cost is \$700.

Hopefully this helps!