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Question 1174253: The unit cost per gizmo is $30,the fixed cost for making gizmos is $1,200. 60 gizmos must be sold in order to break even.
-Find linear cost function
-Find linear revenue function
-Find linear profit function
Show work with all steps
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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