A baseball team plays in a stadium that holds 51,000 spectators. With ticket prices at $10, the average attendance had been 49,000. When ticket prices were lowered to $8, the average attendance rose to 51,000
a. Find the demand function (price p as a function of attendance x), assuming it to be linear
You have two points on the graph:
(10,49) and (8,51)
price went down 2
attendance went up 2
So, the slope is -1, making the demand function
p-49 = -1(x-10)
or
p(x) = 59-x
where x is measured in thousands
To find the demand function, we need to express the price (p) as a function of the attendance (x). Since we're assuming a linear relationship, we can use the slope-intercept form of a linear equation, which is:
y = mx + b
Here, y represents the price (p), x represents the attendance, m represents the slope, and b represents the y-intercept.
We have two data points: (49,000, $10) and (51,000, $8).
Using the first data point, we can substitute the values into the equation to get:
10 = m(49,000) + b
Using the second data point, we can substitute the values into the equation to get:
8 = m(51,000) + b
We now have a system of two equations with two variables. We can solve this system to find the values of m and b.
Multiplying the first equation by -1, we get:
-10 = -49,000m - b
Next, we can add this equation to the second equation to eliminate b:
-10 + 8 = -49,000m - b + m(51,000) + b
-2 = 2,000m
Dividing both sides by 2,000, we find:
m = -1/1,000
Substituting this value of m into the first equation, we get:
10 = (-1/1,000)(49,000) + b
Simplifying, we have:
10 = -49 + b
Adding 49 to both sides:
b = 59
Therefore, the demand function (price p as a function of attendance x) is:
p(x) = (-1/1,000)x + 59
To find the demand function, we need to determine the relationship between the price (p) and the attendance (x). Since we know that the relationship is linear, we can use the formula for a line:
y = mx + b
Where y represents the price (p), x represents the attendance, m represents the slope, and b represents the y-intercept.
Given that the average attendance is 49,000 when the ticket price is $10 and the average attendance is 51,000 when the ticket price is lowered to $8, we can form two data points: (49000, 10) and (51000, 8).
We can now calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Using the values from our data points:
m = (8 - 10) / (51000 - 49000)
m = -2 / 2000
m = -0.001
Now that we have the slope, we can substitute the values of one of the data points into the equation to solve for the y-intercept (b). Let's use the first data point (49000, 10):
10 = (-0.001)(49000) + b
Solving this equation:
10 = -49 + b
b = 10 + 49
b = 59
Therefore, the demand function is given by:
p(x) = -0.001x + 59
This equation represents the relationship between the price (p) and the attendance (x) for the baseball team in the stadium.