Back in 2005, the monorail in a certain city charged $2 per ride and had an average ridership of about 29,000 per day. In December 2005, the monorail company raised the fare to $4 per ride, and average ridership in 2006 plunged to around 15,000 per day. (a) Use the given information to find a linear demand equation. (p is the cost in dollars.) q(p) =

To find the linear demand equation, we can use the point-slope formula, which is: \[ \frac{{\Delta q}}{{\Delta p}} = \frac{{q_2 - q_1}}{{p_2 - p_1}} \] Where: - \( q_1 \) and \( p_1 \) are the initial quantity and price respectively. - \( q_2 \) and \( p_2 \) are the new quantity and price respectively. Given: - \( q_1 = 29000 \) (initial ridership) - \( p_1 = 2 \) (initial fare) - \( q_2 = 15000 \) (new ridership) - \( p_2 = 4 \) (new fare) Plugging in the values: \[ \frac{{15000 - 29000}}{{4 - 2}} \] \[ \frac{{-14000}}{{2}} \] \[ -7000 \] So, the slope of the linear demand equation is -7000. Now, we can use the point-slope formula to find the equation: [ q - q_1 = m(p - p_1) ] [ q - 29000 = -7000(p - 2) ] [ q - 29000 = -7000p + 14000 ] [ q = -7000p + 43000 ] Therefore, the linear demand equation is: [ q(p) = -7000p + 43000 ]