Question

PELO AMOR DE DEUS GENTE EU NÃO ENTENDI NADA ME AJUDEEEEMMMMMMM

Answer 1

$\mathrm{O\ custo\ semanal\ \acute{e}\ dado\ por}\ C(x)=0.1x^2+400.$

$\mathrm{A\ venda\ semanal\ \acute{e}\ dada\ por}\ V(x)=-0.12x^2+30x.$

$\mathbf{a)}$

$\mathrm{O\ lucro\ semanal\ ser\acute{a}\ dado\ por}\text{:}$

$\boxed{L(x)=V(x)-C(x)}$

$\Longrightarrow L(x)=-0.12x^2+30x-(0.1x^2+400)$

$\Longrightarrow \boxed{L(x)=-0.22x^2+30x-400}\ \blacksquare.$

$\mathbf{b)}$

$\mathrm{V\acute{e}rtice}\Longrightarrow \boxed{V(x_v,y_v)=V\bigg(-\dfrac{b}{2a},c-\dfrac{b^2}{4a}\bigg)}$

$\Longrightarrow x_v=-\dfrac{b}{2a}=\dfrac{30}{2(-0.22)}\Longrightarrow x_v=\dfrac{3000}{44}=68.\overline{18}$

$\Longrightarrow y_v=c-\dfrac{b^2}{4a}=-400-\dfrac{30^2}{4(-0.22)}\Longrightarrow y_v=\dfrac{6850}{11}=622.\overline{72}$

$\Longrightarrow \boxed{V\bigg(\dfrac{3000}{44},\dfrac{6850}{11}\bigg)}$

$\mathrm{Intersec\c{c}\tilde{a}o\ com\ o\ eixo}\ x\Longrightarrow \boxed{L(x)=0}$

$\Longrightarrow x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-30\pm\sqrt{548}}{2(-0.22)}=\dfrac{15\mp\sqrt{137}}{0.22}$

$\Longrightarrow x\approx 14.98\ \text{ou}\ x\approx 121.4$

$\Longrightarrow\boxed{(14.98,0)\ \text{e}\ (121.4,0)}$

$\mathrm{Intersec\c{c}\tilde{a}o\ com\ o\ eixo}\ y\Longrightarrow \boxed{x=0}$

$\Longrightarrow L(0)=-0.22(0)^2+30(0)-400\Longrightarrow L(0)=-400$

$\Longrightarrow \boxed{(0,-400)}$

$\mathbf{c)}$

$\mathrm{Os\ pontos\ de\ intersec\c{c}\tilde{a}o\ entre}\ L(x)\ \mathrm{e\ o\ eixo}\ x\ \mathrm{representam\ as}\\ \mathrm{situa\c{c}\tilde{o}es\ onde\ o\ lucro\ \acute{e}\ nulo,\ i.e.,\ quando}\ L(x)=0.$

$\mathbf{d)}$

$\mathrm{O\ eixo\ de\ simetria\ de\ uma\ fun\c{c}\tilde{a}o\ quadr\acute{a}tica\ \acute{e}\ dado\ por}\ x=x_v.$

$x=x_v\Longrightarrow x-x_v=0\Longrightarrow x-\dfrac{3000}{44}=0\Longrightarrow\boxed{44x-3000=0}$

$\mathrm{A\ intersec\c{c}\tilde{a}o\ entre\ a\ par\acute{a}bola\ e\ seu\ eixo\ de\ simetria\ \acute{e}\ o\ v\acute{e}rtice,}\\ \mathrm{que\ demonstra\ o\ valor\ m\acute{a}ximo\ de\ L(x),\ i.e.,\ o\ lucro\ m\acute{a}ximo.}$