Question
A small bicycle manufacturer has daily fixed costs of $1960 and each bicycle costs $73 to manufacture. Let x represent the number of bicycles manufactured and Upper C left parenthesis x right parenthesis represents the cost of manufacturing. Complete parts (a) through (c).
Answers
GPT-4o mini
To address the problem systematically, we will first derive the function that represents the total cost of manufacturing bicycles. Given the information provided, we have two components to the cost:
1. **Fixed costs:** These are constant costs that do not change with the level of production. In this case, the fixed costs are $1960 per day.
2. **Variable costs:** These are costs that change with the level of production. In this case, it costs $73 to manufacture each bicycle.
### Part (a): Write the cost function \( C(x) \)
The total cost function can be expressed as:
\[
C(x) = \text{Fixed Costs} + \text{Variable Costs}
\]
Here, the variable costs depend on the number of bicycles manufactured, which is represented by \( x \). Therefore, the variable cost for \( x \) bicycles is \( 73x \).
Putting this all together, the cost function is:
\[
C(x) = 1960 + 73x
\]
### Part (b): Determine the cost of manufacturing 50 bicycles
To find the cost of manufacturing 50 bicycles, substitute \( x = 50 \) into the cost function \( C(x) \):
\[
C(50) = 1960 + 73(50)
\]
\[
C(50) = 1960 + 3650
\]
\[
C(50) = 5610
\]
So, the cost of manufacturing 50 bicycles is **$5610**.
### Part (c): Determine the number of bicycles that can be manufactured if the total budget is $5000
To find how many bicycles can be manufactured within a budget of $5000, we set the cost function equal to $5000 and solve for \( x \):
\[
C(x) = 5000
\]
\[
1960 + 73x = 5000
\]
Now, isolate \( x \):
\[
73x = 5000 - 1960
\]
\[
73x = 3040
\]
\[
x = \frac{3040}{73} \approx 41.68
\]
Since we cannot manufacture a fraction of a bicycle, we round down to the nearest whole number. Therefore, the maximum number of bicycles that can be manufactured within the budget of $5000 is **41 bicycles**.
### Summary:
- **Cost function**: \( C(x) = 1960 + 73x \)
- **Cost of 50 bicycles**: $5610
- **Maximum bicycles with $5000 budget**: 41 bicycles
1. **Fixed costs:** These are constant costs that do not change with the level of production. In this case, the fixed costs are $1960 per day.
2. **Variable costs:** These are costs that change with the level of production. In this case, it costs $73 to manufacture each bicycle.
### Part (a): Write the cost function \( C(x) \)
The total cost function can be expressed as:
\[
C(x) = \text{Fixed Costs} + \text{Variable Costs}
\]
Here, the variable costs depend on the number of bicycles manufactured, which is represented by \( x \). Therefore, the variable cost for \( x \) bicycles is \( 73x \).
Putting this all together, the cost function is:
\[
C(x) = 1960 + 73x
\]
### Part (b): Determine the cost of manufacturing 50 bicycles
To find the cost of manufacturing 50 bicycles, substitute \( x = 50 \) into the cost function \( C(x) \):
\[
C(50) = 1960 + 73(50)
\]
\[
C(50) = 1960 + 3650
\]
\[
C(50) = 5610
\]
So, the cost of manufacturing 50 bicycles is **$5610**.
### Part (c): Determine the number of bicycles that can be manufactured if the total budget is $5000
To find how many bicycles can be manufactured within a budget of $5000, we set the cost function equal to $5000 and solve for \( x \):
\[
C(x) = 5000
\]
\[
1960 + 73x = 5000
\]
Now, isolate \( x \):
\[
73x = 5000 - 1960
\]
\[
73x = 3040
\]
\[
x = \frac{3040}{73} \approx 41.68
\]
Since we cannot manufacture a fraction of a bicycle, we round down to the nearest whole number. Therefore, the maximum number of bicycles that can be manufactured within the budget of $5000 is **41 bicycles**.
### Summary:
- **Cost function**: \( C(x) = 1960 + 73x \)
- **Cost of 50 bicycles**: $5610
- **Maximum bicycles with $5000 budget**: 41 bicycles