Economists use a cumulative distribution called a Lorenz curve to describe the distribution of income between households in a given country. Typically, a Lorenz curve is defined on the $x$-interval [0, 1] with endpoints (0, 0) and (1, 1), and is continuous, increasing, and concave upward. The points on this curve are determined by ranking all households by income and then computing the percentage of households whose income is less than or equal to a given percentage of the total income of the country. For example, the point (a/100, b/100) is on the Lorenz curve if the bottom $a$ % of the households receive less than or equal to $b$ % of the total income. Absolute equality of income distribution would occur if the bottom $a$ % of the households receive $a$ % of the income, in which case the Lorenz curve would be the line $y = x$. The area between the Lorenz curve and the line $y = x$ measures how much the income distribution differs from absolute equality. The coefficient of inequality is the ratio of the area between the Lorenz curve and the line $y = x$ to the area under $y = x$. As it turns out, one could prove that the coefficient of inequality is twice the area between the Lorenz curve and the line $y = x$, that is $$\textrm{coefficient of inequality } = 2 \int_{0}^{\,1} x-L(x)\, dx.$$

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Suppose that the income distribution for a certain country is represented by the Lorenz curve defined by the equation $$L(x)=\frac{5}{12}x^2+\frac{7}{12}x.$$

(a) What is the percentage of total income received by the bottom 50 % of the households?

Percentage of total income = %

(b) Find the coefficient of inequality.

Coefficient of inequality =