Ex: Determine a Linear Demand Function

Mathispower4u
15 Jul 201305:00
EducationalLearning
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TLDRThe script outlines a method to determine a linear demand function for a baseball team's stadium, given two scenarios with different ticket prices and attendance numbers. It explains the process of finding the slope of the demand curve using the point-slope form, and then deriving the demand function D(Q) with Q as the number of spectators. The final step involves simplifying the equation to a slope-intercept form, providing a clear and concise mathematical model for understanding the relationship between ticket price and attendance.

Takeaways
  • 🏟️ The stadium has a capacity of 66,000 spectators.
  • 🎟️ The initial ticket price was $9.00, with an average attendance of 28,000 spectators.
  • πŸ“‰ When the ticket price was reduced to $6.00, the average attendance increased to 33,000 spectators.
  • πŸ“š The demand function D(Q) is assumed to be linear, where Q represents the number of spectators.
  • πŸ“ˆ The slope-intercept form of the demand function is D(Q) = MxQ + B, with M being the slope and B the intercept.
  • πŸ” To find the slope, calculate the change in price (Ξ”P) divided by the change in quantity (Ξ”Q), resulting in a slope of -3/5,000.
  • πŸ“Œ The ordered pairs for the demand function are (28,000, 9) and (33,000, 6), with Q as the first coordinate and P as the second.
  • πŸ“ Using the point-slope form of the line, the equation is derived as D(Q) - 9 = (-3/5,000)(Q - 28,000).
  • πŸ”’ Simplifying the equation by adding 9 to both sides gives D(Q) = (-3/5,000)Q + 16.8.
  • πŸ“ Converting 16.8 to a fraction with a common denominator results in D(Q) = (-3/5,000)Q + 129/5.
  • πŸ“‰ The final demand function in slope-intercept form is D(Q) = -3/5,000Q + 129/5, which represents the relationship between ticket price and attendance.
Q & A

  • What is the maximum capacity of the stadium mentioned in the script?

    -The maximum capacity of the stadium is 66,000 spectators.

  • What was the initial ticket price and the average attendance before the price change?

    -The initial ticket price was $9.00, and the average attendance was 28,000 spectators.

  • What was the ticket price after it dropped, and how did the average attendance change?

    -The ticket price dropped to $6.00, and the average attendance rose to 33,000 spectators.

  • What type of function is assumed for D of Q in the script?

    -It is assumed that D of Q is a linear function.

  • What is the general form of a linear function in the context of the script?

    -The general form of a linear function is D of Q = M * Q + B, where M is the slope and B is the y-intercept.

  • How are the ordered pairs represented in the context of the demand function?

    -The ordered pairs are represented as (Q, P), where Q is the number of spectators and P is the price.

  • What is the slope of the demand function calculated in the script?

    -The slope of the demand function is calculated to be -3/5000.

  • How is the slope calculated using the two given points in the script?

    -The slope is calculated as the change in price (6 - 9) divided by the change in attendance (33,000 - 28,000).

  • What is the point-slope form of the demand function equation used in the script?

    -The point-slope form used is D of Q - P1 = slope * (Q - Q1), where P1 and Q1 are the price and quantity from one of the given points.

  • What is the final form of the demand function after solving for D of Q in the script?

    -The final form of the demand function is D of Q = (-3/5000) * Q + 129/5.

  • What mathematical operation is suggested to simplify the demand function further in the script?

    -The script suggests distributing the slope fraction and combining like terms to simplify the demand function.

Outlines
00:00
πŸ“‰ Price Elasticity of Stadium Attendance

The script discusses the relationship between ticket prices and stadium attendance for a baseball team. It presents a scenario where a stadium with a capacity of 66,000 has an average attendance of 28,000 when tickets are priced at $9. When the ticket price is reduced to $6, attendance increases to 33,000. The goal is to find a linear demand function D(Q), where Q is the number of spectators. The script explains the process of determining the slope of the demand curve using the change in price and attendance, resulting in a slope of -3/5000. It then uses the point-slope form of a line to derive the demand function, adjusting for the correct order of coordinates where Q is the first element of the pair. The script concludes with a step-by-step guide to solving for D(Q), including algebraic manipulation to express the function in slope-intercept form, resulting in D(Q) = (-3/5000)Q + 129/5.

Mindmap
Keywords
πŸ’‘Baseball Team
A baseball team is a group of players who compete together in the sport of baseball. In the context of the video, the baseball team is central to the discussion as it plays in a stadium with a specific seating capacity, and the video explores the relationship between ticket prices and attendance at their games.
πŸ’‘Stadium
A stadium is a large venue for outdoor sports, concerts, or other events. In the video, the stadium's capacity of 66,000 spectators is a key factor in understanding the potential demand for the baseball team's games.
πŸ’‘Spectators
Spectators are individuals who attend an event to watch it, in this case, a baseball game. The script discusses the number of spectators as a measure of demand, which changes in response to ticket prices.
πŸ’‘Ticket Price
The ticket price is the cost for spectators to attend a baseball game. The video script examines how changes in ticket prices from $9.00 to $6.00 affect the average attendance, illustrating the concept of price elasticity of demand.
πŸ’‘Average Attendance
Average attendance refers to the mean number of spectators present at games over a certain period. The script uses two different average attendance figures (28,000 and 33,000) to demonstrate the impact of ticket price changes.
πŸ’‘Demand Function
A demand function, denoted as D of Q in the script, is a mathematical representation of the relationship between the quantity demanded of a good or service and its price. The video aims to find this function in a linear form to understand how attendance varies with ticket prices.
πŸ’‘Linear
Linear in this context refers to a straight-line relationship between two variables. The script assumes that the demand function is linear, which simplifies the mathematical modeling process and allows for the use of basic algebra to find the demand equation.
πŸ’‘Slope
Slope is a measure of the steepness of a line in a graph, indicating the rate of change of the dependent variable with respect to the independent variable. The script calculates the slope of the demand function to quantify the sensitivity of attendance to price changes.
πŸ’‘Point-Slope Form
The point-slope form is a method of writing the equation of a line using a point on the line and its slope. The video script uses this form to derive the demand function after calculating the slope from the given data points.
πŸ’‘Quantity (Q)
In the context of the demand function, quantity (Q) represents the number of spectators or the number of units of a product that consumers are willing to purchase. The script uses Q to illustrate how the number of attendees changes with the price of tickets.
πŸ’‘Price (P)
Price (P) in the script refers to the cost of a ticket to a baseball game. It is the variable that, along with quantity, defines the demand function and is used to explore how changes in price affect the number of spectators.
Highlights

A baseball team plays in a stadium with a capacity of 66,000 spectators.

The initial ticket price is $9.00, with an average attendance of 28,000 spectators.

When the ticket price is reduced to $6.00, average attendance increases to 33,000 spectators.

The demand function D(Q) is to be found, where Q represents the number of spectators.

The demand function D(Q) is assumed to be linear.

The equation of D(Q) is sought in the slope-intercept form, Y = MX + B.

The function is expressed as D(Q) = M x Q + B, with M as the slope and B as the intercept.

Two data points are given: (28,000, $9.00) and (33,000, $6.00), with Q as the first coordinate and P as the second.

The slope of the demand function is calculated using the change in price and change in attendance.

The slope is determined to be -3 divided by 5,000.

The point-slope form of the line is used to find the demand function D(Q).

The equation is manipulated to solve for D(Q), starting with adding 9 to both sides.

The demand function is expressed in a form that includes the slope and intercept.

Further steps involve distributing the fraction and combining like terms.

The final demand function is simplified to D(Q) = (-3/5000)Q + 129/5.

The process demonstrates how to derive a linear demand function from given data points.

The transcript provides a step-by-step guide to finding the demand function using algebraic methods.

Transcripts

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Demand FunctionStadium EconomicsTicket PricingLinear RegressionSlope InterceptQuantity AnalysisAttendance TrendsPrice ElasticitySports MarketingData Analysis