Question

you invest in a new play. the cost includes an overhead of $27500, plus production costs of $2500 per performance. a sold - out performance brings in $3125 (in solving this exercise, let x represent the number of sold - out performances.)
a. write the cost function, c.
b. write the revenue function, r.
c. determine the break - even point. describe what this means.
c(x)=27500 + 2500x
(type an expression using x as the variable.)
b. write the revenue function, r.
r(x)=3125x
(type an expression using x as the variable.)
c. determine the break - even point.
(44,137500)
(type an ordered pair. do not use commas in the individual coordinates.)
describe what this means.
a. the point where the cost and revenue are equal.
b. the point where the cost and overhead cost are equal.
c. the point where revenue and production cost are equal.
d. the point where revenue and overhead cost are equal.

Explanation:

Step1: Write the revenue function

Revenue is the amount of money generated from sales. If each performance brings in $3125, and \(x\) is the number of sold - out performances, then the revenue function \(R(x)=3125x\).

Step2: Write the cost function

The cost has two components: an overhead cost of $27500 and a production cost of $2500 per performance. Using the formula \(C(x)=\text{fixed cost}+\text{variable cost}\times x\), we get \(C(x)=27500 + 2500x\).

Step3: Find the break - even point

The break - even point occurs when \(R(x)=C(x)\). So we set up the equation \(3125x=27500 + 2500x\).
Subtract \(2500x\) from both sides:
\(3125x-2500x=27500+2500x - 2500x\)
\(625x=27500\)
Divide both sides by 625: \(x=\frac{27500}{625}=44\)
Substitute \(x = 44\) into \(R(x)\): \(R(44)=3125\times44 = 137500\)
So the ordered pair is \((44,137500)\)

Step4: Interpret the break - even point

The break - even point \((44,137500)\) means that when 44 performances are sold out (i.e., \(x = 44\)), the total revenue \(R(44)=137500\) and the total cost \(C(44)=27500+2500\times44=27500 + 110000=137500\). At this point, the cost and revenue are equal.

Answer:

a. \(C(x)=27500 + 2500x\)
b. \(R(x)=3125x\)
c. The break - even point is \((44,137500)\), which means when 44 performances are sold out, the total cost and total revenue are both $137500.