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 of producing tennis rackets is linearly related to the number of tennis rackets produced, we can express this relationship as:
where is the marginal cost per racket, and 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:
Substituting these into the linear equation :
Step 2: Solve for (the marginal cost)
Subtract the first equation from the second to eliminate :
So, the marginal cost per racket is .
Step 3: Solve for (the fixed cost)
Substitute back into one of the original equations, say :
So, the fixed cost is .
Step 4: Write the equation for the total daily cost
The total daily cost function is:
This equation gives the total daily cost of producing 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:
- How much would it cost to produce 70 tennis rackets in a day?
- What is the cost per racket if 100 rackets are produced in a day?
- How many rackets can be produced if the budget is $5,000?
- What is the fixed cost in the context of this problem?
- 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.
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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
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