A jazz group on tour has been drawing average crowds of 600 people. It is projected that for every $1 increase in the $15 ticket price, the average attendance will decrease by 60. At what ticket price will nightly receipts be $7,560?

To find the ticket price at which the nightly receipts will be $7,560, we can set up an equation using the given information. Let's break down the problem step by step.

1. Let's assume the ticket price is x dollars. We are given that for every $1 increase in the ticket price, the average attendance will decrease by 60. This means that if we increase the ticket price by x dollars, the average attendance will decrease by 60x people.

2. Since the initial average attendance is 600 people, if we increase the ticket price by x dollars, the new average attendance will be 600 - 60x people.

3. The nightly receipts can be calculated by multiplying the ticket price by the average attendance. So, the nightly receipts for a ticket price of x dollars will be:
Nightly receipts = x * (600 - 60x)

4. We are given that the nightly receipts should be $7,560. So, we can set up the equation:
7,560 = x * (600 - 60x)

To solve this equation for x, we can follow these steps:

1. Distribute x to both terms inside the parentheses:
7,560 = 600x - 60x^2

2. Rearrange the equation to form a quadratic equation:
60x^2 - 600x + 7,560 = 0

Now, we can either solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the roots. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In the equation 60x^2 - 600x + 7,560 = 0, we have:
a = 60
b = -600
c = 7,560

Substituting these values into the quadratic formula:

x = (-(-600) ± √((-600)^2 - 4 * 60 * 7,560)) / (2 * 60)

Simplifying further:

x = (600 ± √(360,000 - 362,880)) / 120

x = (600 ± √(-2,880)) / 120

Since the term inside the square root is negative, it means there are no real solutions for x. In this case, it suggests that it is not possible to have a ticket price that results in nightly receipts of $7,560.

To find the ticket price at which nightly receipts will be $7,560, we can set up an equation using the given information:

Let's assume the ticket price in dollars as 'x'.
So, the number of tickets sold will be (600 - 60 * (x - 15)).

The total revenue from ticket sales can be calculated by multiplying the number of tickets sold by the ticket price:
Revenue = x * (600 - 60 * (x - 15))

We know that when nightly receipts are $7,560, the revenue will be equal to that amount. So, we can set up the equation:

7560 = x * (600 - 60 * (x - 15))

Let's solve this equation step-by-step:

1. Distribute the operation inside the brackets:
7560 = x * (600 - 60x + 900)

2. Combine like terms:
7560 = x * (-60x + 1500)

3. Distribute x into the parentheses:
7560 = -60x^2 + 1500x

4. Rearrange the equation in standard form:
-60x^2 + 1500x - 7560 = 0

Now we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = -60, b = 1500, and c = -7560.

Let's calculate step-by-step:

1. Calculate the discriminant: √(b^2 - 4ac)
discriminant = √(1500^2 - 4 * (-60) * (-7560))
= √(2250000 - 1814400)
= √(435600)
= 660

2. Calculate the two possible solutions for x:
x1 = (-b + discriminant) / (2a)
= (-1500 + 660) / (2 * (-60))
= (-840) / (-120)
= 7

x2 = (-b - discriminant) / (2a)
= (-1500 - 660) / (2 * (-60))
= (-2160) / (-120)
= 18

Since the context of the question refers to the "increase" in ticket price, we can conclude that the ticket price should be $18 (x = 18) for nightly receipts to be $7,560.