A jazz group on tour has been drawing average crowds of 1000 people. It is projected that for every $1 increase in the $17 ticket price, the average attendance will decrease by 60. What equation could be used to solve this problem and at what ticket price will nightly receipts be $16,400?
let the number of $1 increases be n
number of tickets sold = 1000 - 60n
price per ticket = 17 + n
Receipts = (17+n)(1000-60n)
expand and collect like terms
Set this equal to 16400 and solve using the quadratic formula
To solve this problem, we can use the equation:
Nightly Receipts = Ticket Price * Average Attendance
Let's denote the ticket price as 'x', and the average attendance as 'y'.
From the given information, we know that the average attendance will decrease by 60 for every $1 increase in the ticket price. So, we can express the average attendance as:
y = 1000 - 60(x - 17)
Now, we can substitute the value of 'y' into the equation for nightly receipts:
Nightly Receipts = x * (1000 - 60(x - 17))
To find the ticket price at which the nightly receipts will be $16,400, we can set Nightly Receipts equal to 16,400 and solve for 'x':
16,400 = x * (1000 - 60(x - 17))
Now, we can solve the equation.
To solve this problem, we can use a linear equation that relates the ticket price and the average attendance. Let's define the variables:
- x = ticket price (in dollars)
- y = average attendance
Based on the given information, we know that when the ticket price is $17, the average attendance is 1000. Also, for every $1 increase in the ticket price, the average attendance decreases by 60.
Using this information, we can set up the equation in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Given that the initial ticket price is $17 and the average attendance is 1000, we have the point (17, 1000).
Since the average attendance decreases by 60 for every $1 increase in the ticket price, the slope (m) can be calculated as -60/1 = -60.
Now we have the equation: y = -60x + b.
We can now use the given point to determine the value of b:
1000 = -60(17) + b
1000 = -1020 + b
b = 1000 + 1020
b = 2020.
Now we have the complete equation: y = -60x + 2020.
To find the ticket price that will result in nightly receipts of $16,400, we can substitute y = 16,400 into the equation and solve for x:
16,400 = -60x + 2020.
Rearranging the equation:
-60x = 16,400 - 2020
-60x = 14,380
Dividing by -60:
x = 14,380 / -60
x ≈ -239.67.
Since negative ticket prices don't make sense in this context, we can ignore the negative value. Therefore, the ticket price at which the nightly receipts will be $16,400 is approximately $239.67.