A musician charges 𝐶 of 𝑥 equal to 64𝑥 plus 20,000 dollars, where 𝑥 is the total number of attendees at the concert. The venue charges 80 dollars per ticket. After how many sold tickets does the venue break even? And what is the value of the total tickets sold at that point?
As the venue charges 80 dollars per ticket and 𝑥 is the number of attendees, the revenue that the venue makes is 80𝑥 dollars. We are also told that the musician charges 64𝑥 plus 20,000 dollars. This is the cost for the venue 𝐶 of 𝑥. There are three possible scenarios here. Firstly, when 𝐶 of 𝑥, the cost, is greater than 𝑅 of 𝑥, the revenue, the venue makes a loss. When 𝐶 of 𝑥 is equal to 𝑅 of 𝑥, the venue will break even. Finally, when the revenue exceeds the cost, when 𝐶 of 𝑥 is less than 𝑅 of 𝑥, the venue will make a profit.
We can solve the problem to find the break-even point algebraically or graphically. We will look at the algebraic method first. At the break-even point, 𝐶 of 𝑥 needs to be equal to 𝑅 of 𝑥. Therefore, 80𝑥 is equal to 64𝑥 plus 20,000. Subtracting 64𝑥 from both sides of this equation gives us 16𝑥 is equal to 20,000. We can then divide both sides of this equation by 16. 20,000 divided by 16 is 1,250. The venue needed to sell 1,250 tickets to break even.
The second part of the question asks us to work out the value of the total tickets sold at this point. This is the revenue 𝑅 of 𝑥. As each ticket costs 80 dollars, we need to multiply 80 by 1,250. This is equal to 100,000. At the break-even point, the value of the total tickets sold is 100,000 dollars. We will now look at how we could’ve solved this problem graphically. We begin by plotting the number of tickets on the 𝑥-axis and the amount in dollars on the 𝑦-axis.
For the revenue, we need to draw the straight line 𝑦 is equal to 80𝑥. This line will have a slope or gradient of 80 and an intercept of zero. 80 multiplied by zero is zero. This means that when no tickets are sold, the revenue will be zero dollars. 80 multiplied by 500 is 40,000. When 500 tickets are sold, the revenue is 40,000 dollars. Likewise, when 1,000 tickets are sold, the revenue will be 80,000 dollars and when 1,500 tickets are sold, the revenue is 120,000 dollars. Joining these points gives us the straight line graph for the revenue. The straight line graph for our cost function is 𝑦 is equal to 64𝑥 plus 20,000. This has a slope or gradient of 64 and a 𝑦-intercept of 20,000.
When 500 tickets are sold, the cost is 52,000 dollars as 64 multiplied by 500 plus 20,000 is 52,000. When 1,000 tickets are sold, the cost is 84,000 as 64 multiplied by 1,000 is 64,000 and adding 20,000 to this gives us 84,000. The blue line on our graph shows the cost function for the venue. The point at which the two lines intersect will be the break-even point. This is equal to 1,250. This confirms that the break-even point occurs when 1,250 tickets are sold.
Reading horizontally across to the 𝑦-axis, we see that the break-even point corresponds to 100,000 dollars. This means that both the revenue and cost at this point are 100,000 dollars. When there are 1,250 attendees, the musician charges 100,000 dollars and the venue makes 100,000 dollars from tickets sold.