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\]Question: Find the vertex of the parabola modeling the growth rate of a climate-dependent species, given by $ y = -2x^2 + 8x - 5 $.
Solution: The vertex form of a parabola $ y = ax^2 + bx + c $ has its vertex at $ x = -\frac{b}{2a} $. Here, $ a = -2 $, $ b = 8 $, so $ x = -\frac{8}{2(-2)} = 2 $. Substituting $ x = 2 $ into the equation: $ y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3 $. Thus, the vertex is at $ (2, 3) $. \boxed{(2, 3)}
Question: Solve for $ v $ in the equation $ \frac{1}{2}mv^2 = \frac{3}{4}kT $, where $ m $ is mass, $ k $ is Boltzmann’s constant, and $ T $ is temperature, to express $ v $ in terms of $ k $, $ T $, and $ m $.

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