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

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.

(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:

$C(x) = 90x + 1800$

(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.

(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.

(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.

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.