Could you please help me answer this?
Fund Raising. A charity organization wishes to raise at least $12,000 from a movie premiere to be held at a theatre with 800 seats. The ticket prices are to be $20 and $15, with at least 500 tickets to be sold at $20.
a. If x tickets at $20 and y tickets at $15 are sold, what is the system of inequalities involving x and y?
b. Draw a sketch of the graph of the system in part (a) that shows the region of permissible values of x and y.
Sure, I'd be happy to help you with that!
a. To create a system of inequalities involving x and y, we need to consider the given conditions:
- At least 500 tickets should be sold at $20: x ≥ 500 (since x represents the number of tickets sold at $20).
- The total number of tickets sold should be 800: x + y ≤ 800 (since x represents the number of tickets sold at $20 and y represents the number of tickets sold at $15).
- The charity organization wishes to raise at least $12,000: 20x + 15y ≥ 12,000 (since 20 represents the price of tickets sold at $20 and 15 represents the price of tickets sold at $15).
Therefore, the system of inequalities involving x and y is:
x ≥ 500
x + y ≤ 800
20x + 15y ≥ 12,000
b. To draw the graph of the system above, we need to convert each inequality into an equation, then graph their solution regions on a coordinate plane.
For x ≥ 500, draw a vertical line passing through x = 500 on the x-axis and shade the region to the right of the line since it represents values greater than or equal to 500.
For x + y ≤ 800, start with a horizontal line passing through y = 800 on the y-axis. Shade the region below the line since it represents values less than or equal to 800 - x.
For 20x + 15y ≥ 12,000, rearrange the equation to get y ≥ (12,000 - 20x)/15. This represents a linear equation with a negative slope. Plot a few points on the line, connect them, and shade the region above the line since it represents values greater than or equal to the equation.
The point where all three shaded regions overlap represents the permissible values of x and y. Label this region on your graph.
Remember to label the x-axis as the number of tickets sold at $20 (x), and the y-axis as the number of tickets sold at $15 (y).
I hope this explanation helps!
Sure! Let's answer each part step-by-step:
a. To create the system of inequalities, we can break it down into two parts:
1. The total amount raised from $20 tickets (x) should be greater than or equal to $12,000:
20x ≥ 12,000
2. The total amount raised from $15 tickets (y) should be less than or equal to ($12,000 - amount raised from $20 tickets):
15y ≤ 12,000 - 20x
Therefore, the system of inequalities involving x and y is:
20x ≥ 12,000
15y ≤ 12,000 - 20x
b. To sketch the graph of this system, we'll start by plotting the boundary lines for each inequality:
1. 20x ≥ 12,000 can be rewritten as x ≥ 600 (dividing both sides by 20).
2. 15y ≤ 12,000 - 20x can be rewritten as y ≤ (12,000 - 20x) / 15.
Now, choose a pair of values for both x and y as constants and plot the boundary lines on a coordinate plane. In this case, let's choose x = 600 and x = 800.
For x = 600:
- The point (600, 0) lies on the boundary line x = 600.
- Substitute x = 600 into the second inequality: y ≤ (12,000 - 20*600) / 15 = 400. Therefore, y ≤ 400.
For x = 800:
- The point (800, 0) lies on the boundary line x = 800.
- Substitute x = 800 into the second inequality: y ≤ (12,000 - 20*800) / 15 = 0. Therefore, y ≤ 0.
Now, plot these boundary lines on a graph, and shade the region that satisfies the system of inequalities, which is below or on the lines:
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The shaded region below or on the boundary lines represents the permissible values of x and y for the charity organization to raise at least $12,000.
I hope this helps! Let me know if you have any further questions.