Question

you invest in a new play. the cost includes an overhead of $41,250, 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)=□
(type an expression using x as the variable.)
b. write the revenue function, r.
r(x)=□
(type an expression using x as the variable.)
c. determine the break - even point.

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 \(41250\) and the variable cost is \(2500x\). So, \(C(x)=41250 + 2500x\).

Step2: Write the revenue function

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

Step3: Determine the break - even point

The break - even point occurs when \(C(x)=R(x)\).

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When \(x = 66\), \(R(66)=3125\times66 = 206250\) and \(C(66)=41250+2500\times66=41250 + 165000=206250\)

Answer:

a. \(C(x)=41250 + 2500x\)
b. \(R(x)=3125x\)
c. The break - even point is \((66,206250)\). This means that when there are \(66\) sold - out performances, the total cost and the total revenue are equal, and the investment neither makes a profit nor incurs a loss.