Question

you invest in a new play. the cost includes an overhead of $27,500, 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.
a. write the cost function, c.
c(x)=\square
(type an expression using x as the variable.)

Explanation:

Step1: Write the cost function

The cost function \(C(x)\) is composed of fixed cost (overhead) and variable cost (cost per performance). The fixed cost is \(27500\) and the variable cost is \(2500x\). So, \(C(x)=27500 + 2500x\).

Step2: Write the revenue function

The revenue function \(R(x)\) is based on the income per performance. Given that each sold - out performance brings in \(3125\), so \(R(x)=3125x\).

Step3: Determine the break - even point

Set \(C(x)=R(x)\), then \(27500+2500x = 3125x\).
Subtract \(2500x\) from both sides: \(27500=3125x - 2500x\), which simplifies to \(27500 = 625x\).
Solve for \(x\): \(x=\frac{27500}{625}=44\).

The break - even point means that when \(x = 44\) performances are sold - out, the total cost equals the total revenue, and there is no profit or loss.

Answer:

a. \(C(x)=27500 + 2500x\)
b. \(R(x)=3125x\)
c. Break - even point \(x = 44\). It means when 44 sold - out performances occur, total cost equals total revenue (no profit or loss).