Question

handmade t - shirts
nikolai sells handmade t - shirts at a local event and wants to analyze the business’s financial performance.
cost: $c(x)=4x + 68$
revenue: $r(x)=-x^{2}+40x$
where $x$ represents the number of t - shirts sold.
what are the break - even points?
$(0, 0)$ and $(20, 400)$
$(2, 76)$ and $(34, 204)$
$(0, 0)$ and $(34, 204)$
$(2, 76)$ and $(20, 400)$

Explanation:

Step1: Set cost equal to revenue

$C(x) = R(x)$
$4x + 68 = -x^2 + 40x$

Step2: Rearrange into quadratic equation

$x^2 - 36x + 68 = 0$

Step3: Solve quadratic equation

Use quadratic formula $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1$, $b=-36$, $c=68$
$\sqrt{(-36)^2 - 4(1)(68)} = \sqrt{1296 - 272} = \sqrt{1024} = 32$
$x = \frac{36\pm32}{2}$
$x_1 = \frac{36+32}{2}=34$, $x_2 = \frac{36-32}{2}=2$

Step4: Find corresponding y-values

For $x=2$: $C(2)=4(2)+68=76$
For $x=34$: $C(34)=4(34)+68=204$

Answer:

(2, 76) and (34, 204)