A baseball team plays in a stadium that holds 56000 spectators. With the ticket price at $8 the average attendence has been 22000. When the price dropped to $7, the average attendence rose to 28000. Assume that attendence is linearly related to ticket price.
you have two points on the graph: (8,22) and (7,28)
so, if x = price and y = attendance (in thousands),
y-28 = -6(x-7)
restrict x so that 0 <= y <= 56
Then maximize income = x y
To find the linear relationship between ticket price and attendance, we can use the given information. We know that at a ticket price of $8, the average attendance is 22000, and at a ticket price of $7, the average attendance is 28000.
Let's first determine the change in attendance when the ticket price decreases from $8 to $7.
Attendance change = New attendance - Old attendance
Attendance change = 28000 - 22000
Attendance change = 6000
Next, let's determine the change in ticket price.
Ticket price change = New ticket price - Old ticket price
Ticket price change = $7 - $8
Ticket price change = -$1
Now, we can calculate the attendance change per unit of ticket price change.
Attendance change per unit of ticket price change = Attendance change / Ticket price change
Attendance change per unit of ticket price change = 6000 / -1
Attendance change per unit of ticket price change = -6000
This means that for every $1 decrease in ticket price, the attendance increases by 6000.
To find the equation of the linear relationship, we can use the point-slope form of the equation:
y - y1 = m(x - x1)
Let's choose one set of coordinates, either (8, 22000) or (7, 28000), and substitute them into the equation. Since the question states that attendance is linearly related to ticket price, let's choose (8, 22000).
x1 = 8 (ticket price)
y1 = 22000 (attendance)
Using the point-slope form and substituting the values, we have:
y - 22000 = -6000(x - 8)
Simplifying the equation, we get:
y - 22000 = -6000x + 48000
Now, let's rearrange the equation to solve for y (attendance):
y = -6000x + 70000
Therefore, the equation that represents the linear relationship between ticket price and attendance is y = -6000x + 70000.
To determine the relationship between ticket price and attendance, we can use the concept of a linear equation. The general form of a linear equation is y = mx + b, where y represents the dependent variable (attendance in this case), x represents the independent variable (ticket price), m represents the slope of the line, and b represents the y-intercept.
Given that the average attendance when the ticket price is $8 is 22000 and the average attendance when the ticket price is $7 is 28000, we can use these two points to find the equation of the line.
1. Let's first calculate the slope (m) using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Substitute the given values:
m = (28000 - 22000) / (7 - 8)
m = 6000 / (-1)
m = -6000
2. Now that we have the slope, we can use one of the given points to find the y-intercept (b). Let's use the point (8, 22000):
22000 = -6000(8) + b
22000 = -48000 + b
b = 22000 + 48000
b = 70000
3. Now, we have the values for m and b. The equation for the line representing the relationship between ticket price (x) and attendance (y) is:
y = -6000x + 70000
Using this equation, we can answer questions like "What would be the average attendance if the ticket price is $10?"
To find the attendance when the ticket price is $10, substitute x = 10 into the equation:
y = -6000(10) + 70000
y = -60000 + 70000
y = 10000
Therefore, the average attendance would be 10000 when the ticket price is $10.
By using the linear equation, we can calculate attendance for any ticket price within the given range.