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.

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

\[
C(x) = mx + b
\]

where $ m $ is the marginal cost per racket, and $ b $ 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) = 3855 $
2. $ C(60) = 4245 $

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

\[
3855 = 50m + b
\]
\[
4245 = 60m + b
\]

### Step 2: Solve for $ m $ (the marginal cost)
Subtract the first equation from the second to eliminate $ b $:

\[
4245 - 3855 = (60m + b) - (50m + b)
\]
\[
390 = 10m
\]
\[
m = 39
\]

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

### Step 3: Solve for $ b $ (the fixed cost)
Substitute $ m = 39 $ back into one of the original equations, say $ 3855 = 50m + b $:

\[
3855 = 50(39) + b
\]
\[
3855 = 1950 + b
\]
\[
b = 1905
\]

So, the fixed cost $ b $ is $ 1905 $.

### Step 4: Write the equation for the total daily cost
The total daily cost function is:

\[
C(x) = 39x + 1905
\]

This equation gives the total daily cost $ C(x) $ of producing $ x $ 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.

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. Assuming that daily cost and production are linearly related, find the total daily cost of producing x tennis rackets.

Learn how to calculate the total daily cost of producing tennis rackets using a linear cost function. Given production levels of 50 and 60 rackets, this step-by-step guide explores the relationship between cost and production. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation