Question

handmade necklaces
antonia sells handmade necklaces at a craft fair and needs to assess
the costs and revenue.
cost: $c(x) = 6x + 405$
revenue: $r(x) = -x^2 + 60x$
where $x$ represents the number of necklaces sold.
what are the break - even points?
$(9, 459)$ and $(30, 900)$
$(0, 0)$ and $(30, 900)$
$(0, 0)$ and $(45, 675)$
$(9, 459)$ and $(45, 675)$

Explanation:

Step1: Set cost equal to revenue

$C(x) = R(x)$
$6x + 405 = -x^2 + 60x$

Step2: Rearrange to standard quadratic form

$x^2 - 54x + 405 = 0$

Step3: Solve the quadratic equation

Factor: $(x-9)(x-45)=0$
Solutions: $x=9$ and $x=45$

Step4: Find corresponding y-values

For $x=9$: $C(9)=6(9)+405=459$
For $x=45$: $C(45)=6(45)+405=675$

Answer:

(9, 459) and (45, 675)