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question:Determine the y-intercept of the line represented by the equation -2x + 4y = 5.
answer:To find the y-intercept, we set x = 0 in the equation: {eq}begin{align*} -2(0) + 4y &= 5 4y &= 5 y &= frac{5}{4} end{align*} {/eq} Therefore, the y-intercept is (0, 5/4).
question:Calculate the molarity of a potassium chloride (KCl) solution created by dissolving 0.56 g of KCl in 500 mL of solution.
answer:To calculate the molarity of a solution, we need to know: * The number of moles of solute (KCl) * The volume of the solution in liters Calculating the number of moles of KCl: * Molar mass of KCl = 74.55 g/mol * Number of moles of KCl = mass / molar mass * Number of moles of KCl = 0.56 g / 74.55 g/mol = 0.00751 moles Converting the volume of solution to liters: * Volume of solution = 500 mL * Volume of solution in liters = 500 mL / 1000 mL/L = 0.5 L Calculating the molarity: * Molarity = moles of solute / liters of solution * Molarity = 0.00751 moles / 0.5 L * Molarity = 0.015 M Therefore, the molarity of the potassium chloride solution is 0.015 M.
question:What was the name of the plan proposed by William Paterson that favored the small states, and when was it proposed?
answer:The plan was known as the New Jersey Plan, or the Paterson Plan, which was proposed by William Paterson on June 15, 1787. This plan aimed to counter the Virginia Plan, which advocated for proportional representation based on a state's population, by ensuring equal representation for all states regardless of size.
question:Let f(y,z)=y^3+zy+z^3. Is (0,0) a local min or max? For this I tried using the 2nd derivative test: D=f_{xx}cdot f_{yy}-f_{xy}^2=36xy-1^2 Inserting (0,0): 36cdot 0 cdot 0-1^2=-1<0 Therefore it is a saddle point.
answer:Inserting (0,0): 36cdot 0 cdot 0-1^2=-1<0 Therefore it is a saddle point. Since the second derivative test gives a negative value at (0,0), we can conclude that (0,0) is not a local minimum or maximum. A saddle point is a point on a surface where the function has both positive and negative curvature, so it is neither a local minimum nor a local maximum.