1)The break-even point is the ticket price for which the event's income will equal expenses. Another way to think of the break even point is the ticket price when profit is $0. The equations given in an earlier problem were: P=((4000-25X)x)-7500
I=((4000-25x)x)
S=4000-25x
S=ticket price
x=tickets sold
a. Write and solve an equation that can be used to find the break-even ticket price for this particular planned concert.
b. Write and solve an inequality that can be used to find ticket prices for which the planned concert will make a positive profit.
c. Write and solve an inequality that can be used to find ticket prices for which the planned concert will lose money.
2)What similarities and differences do you see in tables, graphs, and rules of the function relating number of tickets sold, income, and profit to proposed ticket prices?
a. To find the break-even ticket price, we need to set the profit equal to $0. So, the equation would be P = 0. Using the given equation P = ((4000 - 25x)x) - 7500, we can substitute P = 0 and solve for x:
0 = ((4000 - 25x)x) - 7500
Simplify the equation:
0 = 4000x - 25x^2 - 7500
Now, we can solve for x by rearranging the equation to quadratic form:
25x^2 - 4000x + 7500 = 0
We can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula.
b. To find ticket prices for which the planned concert will make a positive profit, we need to set the profit greater than $0. So, the inequality would be P > 0. Using the given equation P = ((4000 - 25x)x) - 7500, we can substitute P as an expression and solve for x:
((4000 - 25x)x) - 7500 > 0
Simplify the inequality:
4000x - 25x^2 - 7500 > 0
Now, we can solve this quadratic inequality by finding the critical points, testing intervals, and determining the sign of the expressions in each interval.
c. To find ticket prices for which the planned concert will lose money, we need to set the profit less than $0. So, the inequality would be P < 0. Using the given equation P = ((4000 - 25x)x) - 7500, we can substitute P as an expression and solve for x:
((4000 - 25x)x) - 7500 < 0
Simplify the inequality:
4000x - 25x^2 - 7500 < 0
Similarly to Part b, we can solve this quadratic inequality by finding the critical points, testing intervals, and determining the sign of the expressions in each interval.
2. Similarities and differences in tables, graphs, and rules of the function relating number of tickets sold, income, and profit to proposed ticket prices:
- Tables: Tables present the values of the number of tickets sold, income, and profit at different ticket prices. They show how these values change as the ticket price varies.
- Graphs: Graphs visually represent the relationship between the number of tickets sold, income, and profit by plotting points and connecting them to form a curve or line. They provide a visual representation of how the values change in relation to each other.
- Rules of the function: The equation given in the problem represents the relationship between the number of tickets sold, income, and profit. This equation serves as a rule for calculating the income and profit based on the number of tickets sold and the ticket price.
The similarities in all these representations are that they all show the relationship between ticket price, number of tickets sold, income, and profit. They provide a way to analyze and understand how changes in ticket price affect the event's financial outcomes. Furthermore, they all provide a way to analyze the break-even point, positive profit, and loss scenarios.
The differences lie in the way information is presented. Tables present the data in a structured tabular format, making it easy to compare values. Graphs provide a visual representation, allowing for an easier understanding of how the variables change. Rules of the function provide an algebraic way to calculate the values based on the given equation.