Question

you invest in a new play. the cost includes an overhead of $37,500, 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

The fixed - cost is $37500$ and the variable cost is $2500$ per performance. So the cost function $C(x)$ is the sum of the fixed cost and the variable cost.
$C(x)=37500 + 2500x$

Step2: Define revenue function

Each sold - out performance brings in $3125$, so the revenue function $R(x)$ is the amount per performance times the number of performances.
$R(x)=3125x$

Step3: Find break - even point

The break - even point occurs when $C(x)=R(x)$. So we set $37500 + 2500x=3125x$.
Subtract $2500x$ from both sides:
$37500=3125x - 2500x$
$37500 = 625x$
Solve for $x$: $x=\frac{37500}{625}=60$.
To find the corresponding $y$ - value, we substitute $x = 60$ into $R(x)$ (we could also use $C(x)$ since they are equal at the break - even point). $R(60)=3125\times60 = 187500$. So the break - even point is $(60187500)$.

The break - even point $(60,187500)$ means that when there are 60 sold - out performances, the total cost and the total revenue are equal, which is $187500$. At this point, the investment in the new play neither makes a profit nor incurs a loss.

Answer:

a. $C(x)=37500 + 2500x$
b. $R(x)=3125x$
c. $(60187500)$