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

Beginner2021-12-21Added 32 answers

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

Beginner2021-12-22Added 39 answers

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:

$x=\frac{-b}{2a}$

$=\frac{-(-10)}{2\times 0.25}$ 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

Expert2021-12-27Added 777 answers

Step 1

The total cost is given by:

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:

the general form for the quadratic equation.

In your problem,

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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