Let x+2y+1=0 and 2x+3y+4=0 are two lines of regression computed from some bivariate data. If \theta is the acute angle between them, then what is the value of 488\tan 3\theta?

Explanation

The slopes of the regression lines are m_1 = -\frac{1}{2} and m_2 = -\frac{2}{3}. The acute angle \theta between them is given by \tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| = \left|\frac{-1/2 - (-2/3)}{1 + 1/3}\right| = \frac{1/6}{4/3} = \frac{1}{8}. Using the identity \tan 3\theta = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}, we get \tan 3\theta = \frac{3(1/8) - (1/8)^3}{1 - 3(1/8)^2} = \frac{191/512}{61/64} = \frac{191}{488}. Thus, 488 \tan 3\theta = 191.

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