A baseball team plays in a stadium that holds 74000 spectators. With the ticket price at $9, the average attendance has been 30000. When the price dropped to $6, the average attendance rose to 37000.
Find the demand function p(x), where x is the number of spectators. (Assume p(x) is linear.)
you have two points (x in thousands)
(9,30) and (6,37)
the slope is -7/3 so
p(x) = -7/3 (x-9) + 30
To find the demand function, we need to determine the relationship between the ticket price and the number of spectators. We know that the demand is linear because the question states that we should assume it is linear.
Let's use the given information to find the equation of the line.
Let x represent the number of spectators and p(x) represent the ticket price.
We have two points on the line: (30000, 9) and (37000, 6).
Using the formula for the equation of a line, we can determine the slope (m) and the y-intercept (b):
m = (y2 - y1) / (x2 - x1)
m = (6 - 9) / (37000 - 30000)
m = -3 / 7000
We can use the point-slope form of the equation to find the y-intercept:
y - y1 = m(x - x1)
y - 9 = (-3 / 7000)(x - 30000)
Since the y-intercept is the value of y when x is zero, we can substitute the values (0, b) into the equation:
b - 9 = (-3 / 7000)(0 - 30000)
b - 9 = 3
Simplifying the equation:
b = 3 + 9
b = 12
Therefore, the equation of the demand function, p(x), is:
p(x) = (-3 / 7000)(x - 30000) + 12
This equation represents the relationship between the ticket price and the number of spectators.
To find the demand function p(x), we need to express the relationship between ticket price and attendance in a linear equation.
Let's use the formula for a linear equation, which is y = mx + b, where y represents the dependent variable (attendance), x represents the independent variable (ticket price), m represents the slope, and b represents the y-intercept.
We have two data points:
1. When the ticket price is $9, the average attendance is 30,000.
2. When the ticket price is $6, the average attendance is 37,000.
So, we can set up two equations using these two data points:
Equation 1: 30,000 = 9m + b
Equation 2: 37,000 = 6m + b
Now, we can solve these two equations simultaneously to find the values of m and b.
To eliminate b (the y-intercept), we can subtract Equation 1 from Equation 2:
(37,000 - 30,000) = (6m + b) - (9m + b)
7,000 = -3m
Rearranging the equation, we get:
-3m = 7,000
m = -7,000/(-3)
m = 2,333.33
Now, substitute the value of m back into either Equation 1 or Equation 2 to find the value of b.
Using Equation 1:
30,000 = 9(2,333.33) + b
30,000 = 20,999.97 + b
30,000 - 20,999.97 = b
9,000.03 = b
Now, we have the slope (m = 2,333.33) and the y-intercept (b = 9,000.03). We can substitute these values into the linear equation formula:
p(x) = mx + b
p(x) = 2,333.33x + 9,000.03
Therefore, the demand function is p(x) = 2,333.33x + 9,000.03, where x is the number of spectators and p(x) is the ticket price.