The problem states that a manufacturer sells belts for $15 per unit. The fixed costs are $3000 per month, and the variable cost per unit is $10. (a) We need to write the equations for the revenue $R(x)$ and cost $C(x)$ functions. (b) We need to find the break-even point.

1. Problem Description

The problem states that a manufacturer sells belts for $15 per unit. The fixed costs are$3000 per month, and the variable cost per unit is $

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0. (a) We need to write the equations for the revenue $R(x)$ and cost $C(x)$ functions.

(b) We need to find the break-even point.

2. Solution Steps

(a)

The revenue function $R(x)$ is the amount of money earned from selling $x$ units, which is the price per unit times the number of units sold. Since the price per unit is $15, the revenue function is

$R(x) = 15x$

The cost function $C(x)$ is the total cost of producing $x$ units, which is the sum of the fixed costs and the variable costs. The fixed costs are $3000, and the variable cost per unit is$10, so the total variable cost is $10x$. Therefore, the cost function is

$C(x) = 3000 + 10x$

(b)

The break-even point is where the revenue equals the cost, so we set $R(x) = C(x)$ and solve for $x$.

$15x = 3000 + 10x$

Subtract $10x$ from both sides:

$15x - 10x = 3000$

$5x = 3000$

Divide by 5:

$x = \frac{3000}{5} = 600$

It takes 600 units to break even.

3. Final Answer

(a) $R(x) = 15x$

$C(x) = 3000 + 10x$

(b) It takes 600 units to break even.