Math Problem Statement

A plant can manufacture 50 tennis rackets per day for a total daily cost of $3,855 and 60 tennis rackets per day for a total daily cost of $4,245.

a. Assuming that daily cost and production are linearly related, find the total daily cost of

producing x tennis rackets.

Solution

Given that the total daily cost C(x)C(x) of producing xx tennis rackets is linearly related to the number of tennis rackets produced, we can express this relationship as:

C(x)=mx+bC(x) = mx + b

where mm is the marginal cost per racket, and bb is the fixed cost (cost when no rackets are produced).

Step 1: Set up the system of equations

We are given two points on the line:

  1. C(50)=3855C(50) = 3855
  2. C(60)=4245C(60) = 4245

Substituting these into the linear equation C(x)=mx+bC(x) = mx + b:

3855=50m+b3855 = 50m + b 4245=60m+b4245 = 60m + b

Step 2: Solve for mm (the marginal cost)

Subtract the first equation from the second to eliminate bb:

42453855=(60m+b)(50m+b)4245 - 3855 = (60m + b) - (50m + b) 390=10m390 = 10m m=39m = 39

So, the marginal cost per racket is m=39m = 39.

Step 3: Solve for bb (the fixed cost)

Substitute m=39m = 39 back into one of the original equations, say 3855=50m+b3855 = 50m + b:

3855=50(39)+b3855 = 50(39) + b 3855=1950+b3855 = 1950 + b b=1905b = 1905

So, the fixed cost bb is 19051905.

Step 4: Write the equation for the total daily cost

The total daily cost function is:

C(x)=39x+1905C(x) = 39x + 1905

This equation gives the total daily cost C(x)C(x) of producing xx tennis rackets.

Would you like details on any specific part of the process or have any questions?


Here are 5 related questions you might consider:

  1. How much would it cost to produce 70 tennis rackets in a day?
  2. What is the cost per racket if 100 rackets are produced in a day?
  3. How many rackets can be produced if the budget is $5,000?
  4. What is the fixed cost in the context of this problem?
  5. How would the cost function change if the marginal cost per racket increased?

Tip: When solving linear cost problems, identifying the fixed and variable components is key to constructing the cost function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Equations
Algebra
Cost Functions

Formulas

C(x) = mx + b (linear cost function)
Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Linear Relationship

Suitable Grade Level

Grades 10-12