 Stefan Hendricks

2021-12-20

A manufacturer of lighting fixtures has daily production costs of $C=800-10x+0.25{x}^{2}.$ where C is the total cost (in dollars) and X is the number of units produced. How many fixtures should be produced each day to yield a minimum cost? einfachmoipf

Equation the function in quadratic form, $f\left(x\right)=a{x}^{2}+bx+c$
$C=0.25{x}^{2}-10x+800$
The function has a minimum when $x=-\frac{b}{2a}$ since $a>0$. To produce a minimum cost, determine the quantity of fixtures that must be manufactured per day. Let $a=0.25$ and $b=-10$
$x=-\frac{b}{2a}$
$x=-\frac{-10}{2\left(0.25\right)}$
$x=-\frac{-10}{0.5}$
$x=20$
The manufacturer should produced 20 lighting fixtures daily to yield a minimum cost. Linda Birchfield

To yield a minimum cost, the number of fixtures that should be produced daily is $x=-\frac{-b}{2a}$. We are aware that $a=0.25$ and $b=-10$
Let's calculate that x's value:
$x=\frac{-b}{2a}$
$=\frac{-\left(-10\right)}{2×0.25}$ We changed a to 0.25 and b to -10.
$=\frac{10}{0.5}$ We multiplied.
$=20$ We divided.
There should be 20 fixtures produced each day. nick1337

Step 1
The total cost is given by:
$y=0.25{x}^{2}-10x+800$
Find the value of x that will give the minimum y.
Notice that the equation is that of a parabola that opens upward. So if you can find the vertex of this parabola, you will have found the minimum. The x-coordinate of the vertex of a parabola can be found by:
$x=\frac{-b}{2×a}$ where the a and b come from:
$a{x}^{2}+bx+c=$
the general form for the quadratic equation.
In your problem, $a=0.25$ and $b=-10$
$x=\frac{-\left(-10\right)}{2×\left(0.25\right)}$ Simplify.
$x=\frac{10}{0.5}$
$x=20$
So 20 fixtures should be produced each day to yield a minimum cost.
If you wanted to find this minimum cost, you would simply substitute x=20 into the original equation and solve for y.

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