GRAPHING PART

$10.$ In this question we are going to model Pareto's Principle:

$(10$ marks$)$

$80\%(0\mathrm{.}8)$ of the work will be done by the $20\%(0\mathrm{.}2)$ of the people who have done the most work.

Let's consider a power function $W=W(p)=a*p^b$ where a and $b$ are parameters to be determined below. The independent variable $p$ represents the fraction $p$ of the population consisting of people that have done the most work, and it takes values between $0$ and $1.$ For example, $p=0\mathrm{.}5$ represents $50\%$ of the whole population, taken from the ones who have done the most work.

$10\mathrm{.}1(2$ marks$)$ What is $W(1)?$ In other words, how much work is done by the entire population?

$W(1)=$

$10\mathrm{.}2(2$ marks$)$ Express Pareto's Principle using the function $W(p).$

$W()=$

$$

$10\mathrm{.}3(3$ marks$)$ Find the parameters a and $b$ of the power function $W(p)=a*p^b$ that satisfies $10\mathrm{.}1$ and $10\mathrm{.}2$ above. Include your calculations below.

$W(p)=$

$10\mathrm{.}4(3$ marks$)$ Sketch a $log-log$ plot for $W(p).$ Label the axes, slope and the intercept.

You can use the space below for your calculations. They won't be marked.