I know I asked this question but I still am getting the wrong answer.
A certain band charges $142 per ticket for concerts when 1918 people attend. The band members realize that for every $3 increase in the ticket price 21 fewer people will attend the concert. Find a revenue function R in terms of x where x represents the number of additional $3 increases in the ticket price.
What should the ticket price be for the band to maximize its revenue?
How many people will attend the concert when the revenue is maximized?
I got the function but I got the ticket price is 22 because that is the critical point of the function am I doing something wrong?
base revenue = 142 * 1918 = 272,356
new revenue R = (142 + 3 x)( 1918-21x)
dR/dx = (142 + 3 x)(-21) + ( 1918-21x)(3) = -2982 -3x + 5754 - 63 x
= 2772 - 66 x
dR/dx = 0 at max or min
x = 2772 / 66 = 42 increases
3 x = 126 amount of increase
142 + 126 = $ 268 per ticket
well, I did that fast but did not get 22
ok but when i enter 22 into the question it tells me that that is correct but it still is saying I'm getting the other 2 questions incorrect
(142 + 3 x)( 1918-21x) = -63x^2 + 2772x + 272356
the vertex occurs at x = -b/2a = 2772/126 = 22
1918-22*21 = 1456 people attended
oh, and x=22 means the ticket price is 142+3*22 = $208
ohhh got it thank you!!
To find the ticket price at which the band maximizes its revenue, we can follow these steps:
Step 1: Determine the equation that relates the ticket price to the number of attendees.
From the given information, we know that for every $3 increase in ticket price, 21 fewer people attend the concert. So we can determine the equation as follows:
x = number of additional $3 increases in the ticket price
1918 - 21x = number of people attending the concert
Step 2: Find the revenue function.
The revenue is calculated by multiplying the ticket price by the number of attendees. So we can write the revenue function as:
R = ticket price * number of attendees
R = (142 + 3x)(1918 - 21x)
Step 3: Simplify the revenue function.
To find the critical point of the revenue function, we need to simplify it. Multiply the terms inside the brackets to simplify the equation:
R = 271976 - 6500x - 443x^2
Step 4: Find the maximum revenue.
To maximize the revenue, we need to find the critical point of the function. This occurs when the derivative of the revenue function is zero.
Taking the derivative of R with respect to x:
R' = -6500 - 886x
Setting R' equal to zero to find the critical point:
-6500 - 886x = 0
-886x = 6500
x = -6500 / -886
x ≈ 7.33 (rounded to two decimal places)
Since x represents the number of additional $3 increases, we can't have a fractional value for x in this context. So we'll consider the nearest whole number, which is 7.
Step 5: Calculate the ticket price for maximum revenue.
Substitute the value of x (7) into the equation to find the ticket price:
Ticket price = 142 + 3x
Ticket price = 142 + 3 * 7
Ticket price = 142 + 21
Ticket price = 163
Therefore, the ticket price at which the band maximizes its revenue is $163.
To determine the number of people who will attend the concert when revenue is maximized, substitute the value of x (7) into the equation we obtained earlier:
Number of people attending = 1918 - 21x
Number of people attending = 1918 - 21 * 7
Number of people attending = 1918 - 147
Number of people attending ≈ 1771
So, when the revenue is maximized, approximately 1771 people will attend the concert.