22.4: Charity Concerts
A popular band is trying to raise at least $20,000 for charity by holding multiple concerts at a park. It plans to sell tickets at $25 each. For each 2-hour concert, the band would need to pay the park $1,250 in fees for security, cleaning, and traffic services.
The band needs to find the combinations of number of tickets sold, t, and number of concerts held, c, that would allow it to reach its fundraising goal.
Create an inequality and graph it on the coordinate plane.
The total amount raised from selling tickets at each concert is given by the equation 25t. Since the band is planning to hold multiple concerts, the total amount raised is also given by 25t * c.
We also know that for each concert, the band will have to pay $1,250 in fees. So the total amount raised after deducting the fees is given by 25t * c - 1,250c.
To reach the fundraising goal of $20,000, the inequality is:
25t * c - 1,250c ≥ 20,000
Graphing this inequality on the coordinate plane would require a 3-D graph, which is not possible to show here. However, we can rewrite the inequality to get a better understanding.
Dividing both sides of the inequality by 25:
tc - 50c ≥ 800
Factoring out c from the left side:
c(t - 50) ≥ 800
Since t and c represent the number of tickets sold and number of concerts held respectively, they must be positive integers. Therefore, we can rewrite the inequality as:
c ≥ 800 / (t - 50)
We can now graph this inequality on a 2-D graph, where t is the x-axis and c is the y-axis.
Graphing the line t = 50 as a vertical line, we can see that the inequality is undefined for t ≤ 50.
For t > 50, the inequality states that c must be greater than or equal to 800 / (t - 50). This means that as t increases, the value of c must also increase to satisfy the inequality. The graph will show a region in the first quadrant where c is greater than or equal to 800 / (t - 50) for t > 50.
Please note that without specific values for t, it is not possible to graph the inequality accurately.
To find the combinations of the number of tickets sold (t) and the number of concerts held (c) that would allow the band to reach its fundraising goal, we can create an inequality based on the given information.
Let's create the inequality:
Total Revenue - Total Expenses ≥ Fundraising Goal
The total revenue can be calculated by multiplying the number of tickets sold by the ticket price, while the total expenses can be calculated by multiplying the number of concerts held by the amount paid to the park for each concert.
Therefore, the inequality is:
t * $25 - c * $1,250 ≥ $20,000
Let's graph this inequality on the coordinate plane in terms of the number of tickets sold (t) on the x-axis and the number of concerts held (c) on the y-axis.
To determine the combinations of tickets sold and concerts held that would allow the band to reach its fundraising goal, we need to create an inequality based on the given information.
Let's denote the number of tickets sold as "t" and the number of concerts held as "c". Each ticket is priced at $25, so the total amount raised from ticket sales would be 25t (25 multiplied by the number of tickets sold).
The band incurs a cost of $1,250 for each concert, regardless of the number of tickets sold. Therefore, the total cost for all concerts would be 1250c (1250 multiplied by the number of concerts held).
In order to reach the fundraising goal of $20,000 or more, the total amount raised from ticket sales minus the total cost for all concerts should be greater than or equal to $20,000. This can be expressed as the following inequality:
25t - 1250c ≥ 20000
Now, let's graph this inequality on the coordinate plane. We can assume positive integer values for t and c since we are dealing with the number of tickets and concerts:
To graph the inequality, first, convert it to slope-intercept form, which is y ≥ mx + b, where y is equivalent to 25t - 1250c, m is the slope, x represents t, and b is the y-intercept:
25t - 1250c ≥ 20000 (subtract 25t from both sides)
-1250c ≥ -25t + 20000 (divide both sides by -1250, be sure to reverse the inequality since we're dividing by a negative number)
c ≤ 0.02t - 16
On the coordinate plane, plot points where c = 0.02t - 16 for different values of t, using integer values for t. Since c should be less than or equal to the expression 0.02t - 16, shade the area below the plotted points.
Note: It is important to limit the range of t and c based on the context of the problem. In this case, positive integer values make sense, but you may need to adjust this based on the specific requirements of the problem.