A baseball team plays in a stadium that holds 68000 spectators. With the ticket price at $8 the average attendence has been 29000. When the price dropped to $5, the average attendence rose to 34000. Assume that attendence is linearly related to ticket price.
You know, I don't mind helping with answers, but don't make me come up with the questions, too!. I assume you want the maximum revenue.
Let x = the number $3-price drops.Then the revenue is
r = (8-3x)(29000+5000x)
Now just find the maximum of r(x), keeping in mind that the stadium only has 68000 seats.
Maybe you are just looking to find the price needed to fill the stadium. But no calculus at all for that.
Your choice!
To solve this problem, we can use the concept of linear equations. Let's define the variables:
Let T represent the ticket price.
Let A represent the average attendance.
We are given the following information:
When the ticket price is $8 (T = 8), the average attendance is 29,000 (A = 29,000).
When the ticket price is $5 (T = 5), the average attendance is 34,000 (A = 34,000).
We can use these points to form two equations and solve for the variables. The equation for a straight line is represented as y = mx + b, where m is the slope and b is the y-intercept.
Equation 1: 29,000 = 8m + b
Equation 2: 34,000 = 5m + b
To find the slope (m), we subtract Equation 2 from Equation 1:
29,000 - 34,000 = 8m - 5m
-5,000 = 3m
m = -5,000 / 3
Now that we have the slope, we can substitute it into one of the equations to find the y-intercept (b). Let's use Equation 1:
29,000 = 8(-5,000 / 3) + b
To simplify this equation, we need to calculate -5,000 / 3:
m ≈ -1,667
Now substitute the calculated value of m:
29,000 = -1,667 + b
To isolate b, subtract -1,667 from both sides:
29,000 + 1,667 = b
b ≈ 30,667
Therefore, the equation that represents the relationship between the ticket price (T) and the average attendance (A) is:
A = (-5,000 / 3)T + 30,667
Now, let's use this equation to answer the question:
To find the attendance when the ticket price is $7 (T = 7), substitute T = 7 into the equation:
A = (-5,000 / 3)(7) + 30,667
A ≈ 29,500
Therefore, the average attendance when the ticket price is $7 is approximately 29,500 spectators.