Question

you are selling handmade bracelets at a local craft fair, and you want to analyze the costs and revenue to understand your business’s financial situation. cost: $c(x) = 2x + 80$ revenue: $r(x) = -x^2 + 26x$ the break - even points are at ▼ and ▼. the maximum revenue occurs when you sell ▼ bracelets.

Explanation:

Step1: Set cost = revenue

$C(x) = R(x)$
$2x + 80 = -x^2 + 26x$

Step2: Rearrange to quadratic equation

$x^2 - 24x + 80 = 0$

Step3: Factor the quadratic

$(x - 4)(x - 20) = 0$

Step4: Solve for x

$x - 4 = 0 \implies x=4$
$x - 20 = 0 \implies x=20$

Step5: Find max revenue x-value

For $R(x)=-x^2+26x$, use vertex formula $x=-\frac{b}{2a}$
$x = -\frac{26}{2(-1)} = 13$

Answer:

The break-even points are at $\boldsymbol{4}$ and $\boldsymbol{20}$.
The maximum revenue occurs when you sell $\boldsymbol{13}$ bracelets.