Question

you invest in a new play. the cost includes an overhead of $40,000, 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) = 40000 + 2500x
(type an expression using x as the variable.)
b. write the revenue function, r.
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: Set up the equation for break - even point

At the break - even point, \(C(x)=R(x)\). So, \(40000 + 2500x=3125x\).

Step2: Solve for \(x\)

Subtract \(2500x\) from both sides: \(40000=3125x - 2500x\).
Simplify the right - hand side: \(40000 = 625x\).
Then, divide both sides by \(625\): \(x=\frac{40000}{625}=64\).

Step3: Find the value of \(R(x)\) (or \(C(x)\)) at \(x = 64\)

Substitute \(x = 64\) into \(R(x)\): \(R(64)=3125\times64 = 200000\).

Answer:

\((64,200000)\)