Questions: A baseball team plays in a stadium that holds 50000 spectators. With the ticket price at 11 the average attendance has been 21000. When the price dropped to 9, the average attendance rose to 25000. Find the demand function p(x), where x is the number of the spectators. (Assume p(x) is linear.) p(x)=

A baseball team plays in a stadium that holds 50000 spectators. With the ticket price at 11 the average attendance has been 21000. When the price dropped to 9, the average attendance rose to 25000.

Find the demand function p(x), where x is the number of the spectators. (Assume p(x) is linear.)
p(x)=

Transcript text: A baseball team plays in a stadium that holds 50000 spectators. With the ticket price at $\$ 11$ the average attendance has been 21000. When the price dropped to $\$ 9$, the averege attendance rose to 25000 . Find the demand function $p(x)$, where $x$ is the number of the spectators. (Assume $p(x)$ is linear.) \[ p(x)= \] $\square$ Submit Question Jump to Answer

Solution

Solution Steps

To find the demand function $ p(x) $, we need to determine the linear relationship between the ticket price and the number of spectators. We have two points: (21000, 11) and (25000, 9). These points represent the number of spectators and the corresponding ticket prices. We can use these points to find the slope of the line and then use the point-slope form to find the equation of the line, which is the demand function.

Step 1: Identify the Given Points

We are given two points that represent the relationship between the number of spectators and the ticket price: $(21000, 11)$ and $(25000, 9)$.

Step 2: Calculate the Slope of the Line

The slope $ m $ of the line connecting these two points is calculated using the formula: \[ m = \frac{p_2 - p_1}{x_2 - x_1} = \frac{9 - 11}{25000 - 21000} = -0.0005 \]

Step 3: Use the Point-Slope Form to Find the Demand Function

Using the point-slope form of a line, $ p(x) = m(x - x_1) + p_1 $, we substitute one of the points and the slope: \[ p(x) = -0.0005(x - 21000) + 11 \]

Step 4: Simplify the Demand Function

Simplifying the equation, we get: \[ p(x) = -0.0005x + 10.5 \]

Step 5: Calculate the Ticket Price for 23000 Spectators

Substitute $ x = 23000 $ into the demand function: \[ p(23000) = -0.0005 \times 23000 + 10.5 = 10.0 \]