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The Penrose method is a system for giving voting powers to members of assemblies or legislatures based on the square root of the number of people that each member represents, divided by 1000. Consider a parliament that represents the people of the world and how voting power might be given to different nations. The table shows the estimated populations of Canada and the three most populous and the three least populous countries in the world. $$\begin{array}{cc} {\text { Country }} & \text { Population } \\\\\hline \text { China } & 1361513000 \\\\\hline \text { India } & 1251696000 \\\\\hline \text { United States } & 325540000 \\\\\hline \text { Canada } & 35100000 \\\\\hline \text { Tuvalu } & 11000 \\\\\hline \text { Nauru } & 10000 \\\\\hline \text { Vatican City } & 1000 \\ \hline\end{array}$$ a) Share your answers to the following two questions with a classmate and explain your thinking: \(\bullet\) Which countries might feel that a "one nation, one vote" system is an unfair way to allocate voting power? \(\bullet\) Which countries might feel that a "one person, one vote" system is unfair? b) What percent of the voting power would each nation listed above have under a "one person, one vote" system, assuming a world population of approximately 7.302 billion? c) If \(x\) represents the population of a country and \(V(x)\) represents its voting power, what function could be written to represent the Penrose method? d) Under the Penrose method, the sum of the world voting power using the given data is approximately \(765 .\) What percent of the voting power would this system give each nation in the table? e) Why might the Penrose method be viewed as a compromise for allocating voting power?

Heron's formula, \(A=\sqrt{s(s-a)(s-b)(s-c)},\) relates the area, \(A\), of a triangle to the lengths of the three sides, \(a, b,\) and \(c,\) and its semi-perimeter (half its perimeter), \(s=\frac{a+b+c}{2} .\) A triangle has an area of \(900 \mathrm{cm}^{2}\) and one side that measures \(60 \mathrm{cm} .\) The other two side lengths are unknown, but one is twice the length of the other. What are the lengths of the three sides of the triangle?

Develop a formula for radius as a function of surface area for a) a cylinder with equal diameter and height b) a cone with height three times its diameter