Question

As dimensões de um retângulo são numericamente iguais às
coordenadas do vértice da parábola de equação
y = - 81x² + 27 x +2. A área do retângulo é:

A=17/24
B=48/9
C=37/3
D=14/19

Answer 1

Boa noite!


Solução!


Para determinarmos as coordenadas do vertice vamos usar essa formula.



$V\left ( \dfrac{-b}{2.a}, -\dfrac{\Delta}{4.a} \right )$



$X_{V}= \dfrac{-b}{2.a}\\\\\\\\\ Y_{V} -\dfrac{\Delta}{4.a}$



$y=-81 x^{2} +27x+22\\\\\\ a=-81\\\\\ b=27\\\\\\ c=2$



$X_{V}= \dfrac{-27}{2.(-81)}\\\\\\ X_{V}= \dfrac{-27}{-162}\\\\\\ X_{V}= \dfrac{-27:27}{-162:27}\\\\\\ X_{V}= \dfrac{1}{6}$




$Y_{V}= -\dfrac{\Delta}{4.a}\\\\\\ Y_{V}= -\dfrac{b^{2}-4.a.c }{4.a}\\\\\\ Y_{V}= -\dfrac{(27)^{2}-4.(-81).2 }{4.(-81)}\\\\\\ Y_{V}= -\dfrac{729+648 }{-324}\\\\\\ Y_{V}= -\dfrac{1377 }{-324}\\\\\\ Y_{V}= \dfrac{1377 }{324}\\\\\\\\ Simplificando~~ por~~81!\\\\\\\ Y_{V}= \dfrac{1377 :81}{324:81}= \dfrac{17}{4}$



A area do retângulo é dada pela formula!


$Area=Base \times Altura~~~~~~~Base= \dfrac{1}{6}~~~~Altura= \dfrac{17}{4}\\\\\\\\ Area= \dfrac{1}{6} \times\dfrac{17}{4}\\\\\\\\ Area= \dfrac{17}{24}\\\\\\\\ Resposta:Alternativa~~A$


Boa noite!
Bons estudos!