Question

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) =

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Murray

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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 ]

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