Question

1. peter’s promoting is organizing a concert. the cost of the venue and the rock band is $15 000. each concert ticket sells for $300. peter’s profit is the money he makes from selling tickets minus the cost. let ( n ) represent the number of tickets sold. let ( p ) represent peter’s profit. an equation that relates the profit to the number of tickets sold is: ( p = 300n - 15 000 ) a) create a table of values for the relation. use these values of ( n ): 10, 20, 30, 40, 50, 60, 70, 80 b) graph the relation. what do negative values of ( p ) represent? c) describe the relationship between the variables in the graph. d) how can you use the graph to find the profit when 75 tickets are sold?

Explanation:

Part (a)

Step1: Substitute \( n = 10 \) into \( p = 300n - 15000 \)

\( p = 300\times10 - 15000 = 3000 - 15000 = -12000 \)

Step2: Substitute \( n = 20 \) into \( p = 300n - 15000 \)

\( p = 300\times20 - 15000 = 6000 - 15000 = -9000 \)

Step3: Substitute \( n = 30 \) into \( p = 300n - 15000 \)

\( p = 300\times30 - 15000 = 9000 - 15000 = -6000 \)

Step4: Substitute \( n = 40 \) into \( p = 300n - 15000 \)

\( p = 300\times40 - 15000 = 12000 - 15000 = -3000 \)

Step5: Substitute \( n = 50 \) into \( p = 300n - 15000 \)

\( p = 300\times50 - 15000 = 15000 - 15000 = 0 \)

Step6: Substitute \( n = 60 \) into \( p = 300n - 15000 \)

\( p = 300\times60 - 15000 = 18000 - 15000 = 3000 \)

Step7: Substitute \( n = 70 \) into \( p = 300n - 15000 \)

\( p = 300\times70 - 15000 = 21000 - 15000 = 6000 \)

Step8: Substitute \( n = 80 \) into \( p = 300n - 15000 \)

\( p = 300\times80 - 15000 = 24000 - 15000 = 9000 \)

The table of values is:

\( n \)	10	20	30	40	50	60	70	80

Part (b)

To graph the relation \( p = 300n - 15000 \), we can use the table of values from part (a). Plot the points \((n, p)\) where \( n \) is on the x - axis and \( p \) is on the y - axis. For example, the points are \((10, - 12000)\), \((20, - 9000)\), \((30, - 6000)\), \((40, - 3000)\), \((50, 0)\), \((60, 3000)\), \((70, 6000)\), \((80, 9000)\). Then draw a straight line through these points since the equation is linear.

Negative values of \( p \) represent a loss. That is, when the value of \( p \) is negative, the money made from selling tickets is less than the cost of the venue and the rock band, so Peter is losing money.

Part (c)

The relationship between the variables \( n \) (number of tickets sold) and \( p \) (profit) is a linear relationship. The equation \( p = 300n-15000 \) is in the form of a linear equation \( y = mx + b \) (where \( y = p \), \( x = n \), \( m = 300 \) and \( b=- 15000 \)). As the number of tickets sold (\( n \)) increases by 1, the profit (\( p \)) increases by 300 (since the slope \( m = 300 \)). Also, when \( n = 50 \), the profit \( p = 0 \), which means that 50 tickets sold is the break - even point (where revenue equals cost).

Part (d)

Answer:

Step1: Locate \( n = 75 \) on the x - axis (the axis representing the number of tickets sold).

Step2: Move vertically until you intersect the graph of the line \( p = 300n - 15000 \).

Step3: Then move horizontally to the y - axis (the axis representing profit) to find the corresponding value of \( p \).

We can also calculate the profit directly using the formula: \( p=300\times75 - 15000=22500 - 15000 = 7500 \). From the graph, we should get the same result.

Final Answers

(a)

\( n \)	10	20	30	40	50	60	70	80

(b)

Graph: Plot the points from the table and draw a straight line. Negative \( p \) values represent a loss (revenue < cost).

(c)

The relationship is linear. As \( n \) (tickets sold) increases by 1, \( p \) (profit) increases by 300. At \( n = 50 \), \( p = 0 \) (break - even).

(d)

Locate \( n = 75 \) on the x - axis, move up to the line, then left to the y - axis to find \( p \). (Direct calculation: \( p = 7500 \))