A baseball team plays in a stadium that holds 58000 spectators. With the ticket price at $11 the average attendance has been 22000. When the price dropped to $10, the average attendance rose to 29000. Assume that attendance is linearly related to ticket price.
is there a question in there somewhere?
Let x be the number of $1 price decreases. Then the attendance is
22000 + 7000x
That means the revenue is
r(x) = (11-x)(22000+7000x) = -1000(7x^2-55x-242)
dr/dx = -1000(14x-55)
maximum revenue occurs at the vertex, when x = 55/14 = 3.92
That is, max revenue of $350,036 is when the price is reduced to $8.08 and the attendance is 49,300
To determine the equation of the line that represents the relationship between the ticket price and the average attendance, we can use the slope-intercept form of a linear equation, which is y = mx + b, where y represents the average attendance, x represents the ticket price, m represents the slope, and b represents the y-intercept.
We are given two data points: (11, 22000) and (10, 29000).
First, let's calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (29000 - 22000) / (10 - 11)
m = 7000 / (-1)
m = -7000
Using the point-slope form of a linear equation, we can substitute one of the points (11, 22000) along with the calculated slope into the equation:
y - y1 = m(x - x1)
y - 22000 = -7000(x - 11)
Next, we will simplify the equation to find the y-intercept:
y - 22000 = -7000x + 77000
y = -7000x + 99000
Therefore, the equation that represents the relationship between the ticket price (x) and the average attendance (y) is:
y = -7000x + 99000.
To determine the relationship between ticket price and attendance, we can use the given information about the average attendance at two different ticket prices.
Let's start by finding the slope of the linear relationship between ticket price and attendance.
The formula for calculating the slope, denoted by "m," is:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Let's assign the first point as (10, 29000), where the ticket price is $10 and the average attendance is 29000.
Now, let's assign the second point as (11, 22000), where the ticket price is $11 and the average attendance is 22000.
Plugging these values into the slope formula, we have:
m = (22000 - 29000) / (11 - 10)
m = -7000
So, the slope of the line is -7000.
Now, let's find the equation of the line using the slope-intercept form: y = mx + b, where "m" is the slope and "b" is the y-intercept.
We know the slope (m = -7000) and need to find the y-intercept (b).
To do this, we can use either of the two given data points:
Using the point (10, 29000), we can substitute the values into the equation as follows:
29000 = -7000(10) + b
29000 = -70000 + b
b = 99000
Now we have the equation: y = -7000x + 99000
The equation describes the relationship between ticket price (x) and attendance (y) for this baseball team.
To find the maximum attendance, we need to determine the ticket price that would yield the highest value of y.
Since the equation is linear, the maximum attendance will occur when the ticket price is at its extreme value, in this case, either the lowest or highest value.
To find the maximum attendance, we will compare the attendance at both ends.
When the ticket price is $10 (x = 10), the attendance is:
y = -7000(10) + 99000
y = 29000
When the ticket price is $11 (x = 11), the attendance is:
y = -7000(11) + 99000
y = 22000
Comparing the two values, we see that the maximum attendance is 29000, which occurs when the ticket price is $10.
Therefore, in order to maximize attendance, the ticket price should be set at $10.