Question

alguém pode me ajudar em mano? por favor​

Answer 1

a)

$W(V) = C\cdot ln\left(\dfrac{V}{V_o}\right)\\\\\\\Rightarrow \dfrac{dW}{dV} = C\cdot \left[ln\left(\dfrac{V}{V_o}\right)\right]' \textsf{--- Regra da cadeia} \\\\\\\Rightarrow \dfrac{dW}{dV} = C\cdot \dfrac{1}{V_o} \cdot \dfrac{1}{V/V_o} ~~ \textsf{---Simplifique o denominador}\\\\\\\Rightarrow \dfrac{dW}{dV} = \dfrac{C}{V}$

$V(t) = 2t^4 + 1\\\\\\ \Rightarrow \dfrac{dV}{dt} = 8t^3~~\textsf{---Regra do "tombo"}$

b)

$\dfrac{d(W(V(t))}{dt} = \dfrac{d(C\cdot ln\left(\frac{2t^4 + 1}{V_o}\right))}{dt}\\\\\\\Rightarrow\dfrac{dW(V(t))}{dt} = C\cdot \left[ln\left(\dfrac{2t^4+1}{V_o}\right)\right]' \textsf{--- Regra da cadeia} \\\\\\\Rightarrow \dfrac{dW(V(t))}{dt} = C\cdot \dfrac{8t^3}{V_o} \cdot \dfrac{1}{(2t^4+1)/V_o} ~~ \textsf{---Simplifique o denominador}\\\\\\\Rightarrow \dfrac{dW(V(t))}{dt} = \dfrac{C\cdot 8t^3}{2t^4+1} \\\\\\ \Rightarrow \dfrac{dW(V(t))}{dt} = C\cdot\dfrac{V'(t)}{V(t)}$

c)

$V(t) = 2t^4 + 1 \\\\33 = 2t^4 + 1 \\\\32 = 2t^4 \\\\t^4 = 16 \Rightarrow \boxed{t = 2}$

$V'(2) = 8.(2)^3 \Rightarrow \boxed{V'(2) = 64}$

$Pot = 10 \cdot \dfrac{V'(t)}{V(t)} = 10 \cdot \dfrac{64}{33} \approx \boxed{\boxed{19{,}33}}$