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.
c(x)=27500 + 2500x
r(x)=3125x
(type an expression using x as the variable.)
c. determine the break - even point.
(type an ordered pair. do not use commas in the individual coordinates.)

Explanation:

Step1: Write cost function

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

Step2: Write revenue function

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

Step3: Find break - even point

At break - even point, \( C(x)=R(x) \).

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Substitute \( x = 44 \) into \( R(x) \) (or \( C(x) \)), \( R(44)=3125\times44=137500 \)

Answer:

a. \( C(x)=27500 + 2500x \)
b. \( R(x)=3125x \)
c. The break - even point is \( (44,137500) \). It means when the number of sold - out performances \( x = 44 \), the total cost is equal to the total revenue, and both are \( 137500 \). At this point, there is neither profit nor loss for the investment in the new play.