Question

you invest in a new play. the cost includes an overhead of $22,500, 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)= (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. (type an ordered pair. do not use commas in the individual coordinates.) describe what this means.

Explanation:

Step1: Define cost function components

The overhead cost is a fixed - cost of $22500$ and the variable cost is $2500$ per performance. The cost function $C(x)$ is the sum of fixed and variable costs. So, $C(x)=22500 + 2500x$.

Step2: Define revenue function

Since each sold - out performance brings in $3125$ and $x$ is the number of sold - out performances, the revenue function $R(x)$ is given by the product of the revenue per performance and the number of performances. So, $R(x)=3125x$.

Step3: Find break - even point

The break - even point occurs when $C(x)=R(x)$. Set $22500 + 2500x=3125x$.
Subtract $2500x$ from both sides: $22500=3125x - 2500x$.
Simplify the right - hand side: $22500 = 625x$.
Solve for $x$: $x=\frac{22500}{625}=36$.
To find the corresponding $y$ - value (revenue or cost at break - even), substitute $x = 36$ into $R(x)$ (or $C(x)$). $R(36)=3125\times36 = 112500$. The break - even point is the ordered pair $(36,112500)$.

Answer:

a. $C(x)=22500 + 2500x$
b. $R(x)=3125x$
c. $(36,112500)$; The break - even point $(36,112500)$ means that when there are 36 sold - out performances, the total cost and the total revenue are both $112500$, and the investment in the new play neither makes a profit nor incurs a loss.