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Let's go through the problem step by step. Here’s what each part of the question asks:

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### 47. Cost Function Problem:

#### Given:
- Fixed costs: $1800 (which corresponds to the y-intercept $b$)
- Cost to manufacture each bicycle: $90 (this is the slope $m$)

The general form of the cost function is $C(x) = mx + b$, where:
- $x$ is the number of bicycles produced,
- $C(x)$ is the total cost,
- $m = 90$, the cost per bicycle,
- $b = 1800$, the fixed cost.

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#### (a) Write a linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day.

We substitute the values for $m$ and $b$ into the linear equation:

\[
C(x) = 90x + 1800
\]

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#### (b) Graph the model.

The graph would be a straight line where:
- The y-intercept is $1800$, meaning when no bicycles are produced, the fixed cost is $1800.
- The slope is $90$, meaning for each additional bicycle, the cost increases by $90.
  
Plot the line starting at (0, 1800) with a slope of 90. The x-axis represents the number of bicycles, and the y-axis represents the total cost.

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#### (c) What is the cost of manufacturing 14 bicycles in a day?

Using the equation $C(x) = 90x + 1800$, substitute $x = 14$:

\[
C(14) = 90(14) + 1800 = 1260 + 1800 = 3060
\]

So, the cost of manufacturing 14 bicycles in a day is **$3060**.

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#### (d) How many bicycles could be manufactured for $3780?

We need to solve the equation $C(x) = 3780$ for $x$:

\[
3780 = 90x + 1800
\]

Subtract 1800 from both sides:

\[
1980 = 90x
\]

Now, divide by 90:

\[
x = \frac{1980}{90} = 22
\]

So, **22 bicycles** could be manufactured for $3780.

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Would you like me to provide more details on any of these steps, or have any questions?

Here are some related questions to expand your understanding:
1. How does the slope of the cost function impact the total cost?
2. What happens to the cost function if the fixed costs increase by $500?
3. Can you calculate the cost of manufacturing 30 bicycles in a day?
4. How would the equation change if each bicycle cost $100 instead of $90?
5. How many bicycles could be manufactured for $5000?

**Tip:** In linear models, the slope tells you the rate of change (in this case, how much each additional bicycle costs), while the intercept represents the fixed costs or starting value.

Suppose that a small bicycle manufacturer has daily fixed costs of $1800, and each bicycle costs $90 to manufacture.
(a) Write a linear model that expresses the cost C of manufacturing x bicycles in a day.
(b) Graph the model.
(c) What is the cost of manufacturing 14 bicycles in a day?
(d) How many bicycles could be manufactured for $3780?

Learn how to formulate and analyze the linear cost function for a bicycle manufacturer with daily fixed costs of $1800 and a variable cost of $90 per bicycle. Find out how to calculate the total manufacturing cost for a given number of bicycles and how to determine the number of bicycles produced for a specific cost.

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