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Assume that the situation can be expressed as a linear cost function. Find the cost function. Fixed cost is $200, 50 items cost $1,700 to produce. The linear cost function is C(x)=

Assume that the situation can be expressed as a linear cost function. Find the cost function. Fixed cost is $200, 50 items cost $1,700 to produce. The linear cost function is C(x)=

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Understanding Linear Cost Functions #### Problem Statement: Assume that the situation can be expressed as a linear cost function. Find the cost function. **Given Data:** - Fixed cost is $200. - The cost to produce 50 items is $1,700. #### Finding the Linear Cost Function: A linear cost function can be expressed in the form: \[ C(x) = mx + b \] where: - \( C(x) \) is the total cost, - \( m \) is the variable cost per unit, - \( x \) is the number of units produced, - \( b \) is the fixed cost. **Step-by-Step Solution:** 1. **Identify the fixed cost \( b \):** It is given that the fixed cost is $200. Thus, \( b = 200 \). 2. **Set up the equation for the total cost of producing 50 items:** Given that producing 50 items costs $1,700, we can set up the equation: \[ C(50) = 50m + 200 = 1700 \] 3. **Solve for \( m \):** \[ 50m + 200 = 1700 \] Subtract 200 from both sides: \[ 50m = 1500 \] Divide by 50: \[ m = 30 \] **Conclusion:** The linear cost function is: \[ C(x) = 30x + 200 \] This means the variable cost per unit is $30, and the fixed cost is $200.
Transcribed Image Text:### Understanding Linear Cost Functions #### Problem Statement: Assume that the situation can be expressed as a linear cost function. Find the cost function. **Given Data:** - Fixed cost is $200. - The cost to produce 50 items is $1,700. #### Finding the Linear Cost Function: A linear cost function can be expressed in the form: \[ C(x) = mx + b \] where: - \( C(x) \) is the total cost, - \( m \) is the variable cost per unit, - \( x \) is the number of units produced, - \( b \) is the fixed cost. **Step-by-Step Solution:** 1. **Identify the fixed cost \( b \):** It is given that the fixed cost is $200. Thus, \( b = 200 \). 2. **Set up the equation for the total cost of producing 50 items:** Given that producing 50 items costs $1,700, we can set up the equation: \[ C(50) = 50m + 200 = 1700 \] 3. **Solve for \( m \):** \[ 50m + 200 = 1700 \] Subtract 200 from both sides: \[ 50m = 1500 \] Divide by 50: \[ m = 30 \] **Conclusion:** The linear cost function is: \[ C(x) = 30x + 200 \] This means the variable cost per unit is $30, and the fixed cost is $200.
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