Question 1174253
Absolutely! Let's break down this problem step-by-step.

**1. Linear Cost Function (C(x))**

* **Fixed Cost:** $1,200
* **Unit Cost:** $30 per gizmo
* **x:** Number of gizmos produced

The linear cost function is the sum of the fixed cost and the variable cost (unit cost multiplied by the number of gizmos):

   * C(x) = Fixed Cost + (Unit Cost * x)
   * C(x) = 1200 + 30x

**2. Linear Revenue Function (R(x))**

* **Break-even Point:** 60 gizmos
* At the break-even point, total revenue equals total cost.

First, let's find the total cost at the break-even point:

   * C(60) = 1200 + 30(60)
   * C(60) = 1200 + 1800
   * C(60) = 3000

Since revenue equals cost at the break-even point:

   * R(60) = 3000

Now, let's find the selling price per gizmo:

   * Selling Price = Total Revenue / Number of Gizmos
   * Selling Price = 3000 / 60
   * Selling Price = 50

Therefore, the linear revenue function is:

   * R(x) = Selling Price * x
   * R(x) = 50x

**3. Linear Profit Function (P(x))**

* Profit is the difference between revenue and cost:

   * P(x) = R(x) - C(x)
   * P(x) = 50x - (1200 + 30x)
   * P(x) = 50x - 1200 - 30x
   * P(x) = 20x - 1200

**Summary**

* **Linear Cost Function:** C(x) = 1200 + 30x
* **Linear Revenue Function:** R(x) = 50x
* **Linear Profit Function:** P(x) = 20x - 1200