Question

a baseball team plays in a stadium that holds 62000 spectators. with the ticket price at $8 the average attendance has been 24000. when the price dropped to $6, the average attendance rose to 31000. find a demand function ( d(q) ), where ( q ) is the quantity/number of the spectators. (assume ( d(q) ) is linear)( d(q) = ) (for best results, keep answers in fraction form, not decimals)

Explanation:

Step1: Identify two points

We have two points \((q_1, p_1)=(24000, 8)\) and \((q_2, p_2)=(31000, 6)\) where \(q\) is quantity and \(p\) is price.

Step2: Calculate the slope \(m\)

The slope formula is \(m=\frac{p_2 - p_1}{q_2 - q_1}\). Substituting values: \(m=\frac{6 - 8}{31000 - 24000}=\frac{-2}{7000}=-\frac{1}{3500}\)

Step3: Use point - slope form

Using point - slope form \(p - p_1=m(q - q_1)\). Substitute \(p_1 = 8\), \(q_1 = 24000\) and \(m=-\frac{1}{3500}\):
\(p - 8=-\frac{1}{3500}(q - 24000)\)

Step4: Solve for \(p = D(q)\)

\(p=-\frac{1}{3500}q+\frac{24000}{3500}+8\)
Simplify \(\frac{24000}{3500}=\frac{48}{7}\) and \(8=\frac{56}{7}\)
\(p=-\frac{1}{3500}q+\frac{48 + 56}{7}=-\frac{1}{3500}q+\frac{104}{7}\)

Answer:

\(D(q)=-\frac{1}{3500}q+\frac{104}{7}\)