Answer:
99 units.
Step-by-step explanation:
The cost function for manufacturing x units of a certain product is:
[tex]C(x)=x^2-15x+34[/tex]
We want to find the number of units manufactured at a cost of $8350. Therefore:
[tex]8350=x^2-15x+34[/tex]
Subtract 8350 from both sides:
[tex]x^2-15x-8316=0[/tex]
This equation can be a bit difficult to factor, if even possible, so we can use the quadratic formula:
[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
In this case, a = 1, b = -15, and c = -8316. Thus:
[tex]\displaystyle x=\frac{-(-15)\pm\sqrt{(-15)^2-4(1)(-8316)}}{2(1)}[/tex]
Simplify:
[tex]\displaystyle x=\frac{15\pm\sqrt{33489}}{2}[/tex]
Evaluate:
[tex]\displaystyle x=\frac{15\pm183}{2}[/tex]
Therefore, our solutions are:
[tex]\displaystyle x=\frac{15+183}{2}=99\text{ and } x=\frac{15-183}{2}=-84[/tex]
We cannot produce negative items, so we can ignore the second answer.
Therefore, for a cost of $8350, 99 items are being produced.
