Let's break this down step by step:

(a) Find the cost function.

The total cost $C(x)$ for producing $x$ sprockets is the sum of the fixed costs and the variable costs.

• Fixed costs: These are constant costs that do not change with the number of sprockets produced, which is $10,000.
• Variable costs: These depend on the number of sprockets produced. Since it costs $50 to produce each sprocket, the variable cost is $50x$, where $x$ is the number of sprockets.

Thus, the total cost function is: $C(x) = 10,000 + 50x$ where:

• $C(x)$ is the total cost to produce $x$ sprockets.
• $10,000$ is the fixed cost.
• $50x$ is the variable cost per sprocket.

(b) Graph the cost function using Desmos.

You can plot this function in Desmos using the following steps:

1. Go to Desmos.
2. In the expression box, enter the cost function: $C(x) = 10,000 + 50x$
3. Adjust the viewing window so that the graph is clearly visible. A good starting window would be:
   • $x$ from 0 to 200 (depending on how many sprockets you want to observe).
   • $y$ from 0 to 15,000.

Once plotted, you can save the graph.

(c) Is the graph increasing or decreasing?

The cost function $C(x) = 10,000 + 50x$ is a straight line with a positive slope of 50. This means that as $x$ increases (i.e., as more sprockets are produced), the cost increases. Therefore, the graph is increasing.

(d) Find the y-intercept.

The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the cost function: $C(0) = 10,000 + 50(0) = 10,000$ So, the y-intercept is $10,000$.

(e) Interpret the y-intercept in the context of this problem.

The y-intercept represents the total cost when no sprockets are produced ($x = 0$). In this context, it means that the company has fixed costs of $10,000, even if they do not produce any sprockets. These costs could include things like equipment maintenance, rent, and salaries for staff that are not directly tied to production.

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Would you like more details on any part of this problem? Here are some related questions you might find interesting:

1. How would the cost function change if the fixed costs were higher?
2. What happens to the cost per sprocket if the production cost per unit increases?
3. How can you calculate the total cost for producing 100 sprockets?
4. What is the slope of the cost function, and what does it represent?
5. If the company produces 200 sprockets, how much total cost will it incur?

Tip: When analyzing cost functions, the slope (the coefficient of $x$) represents the variable cost per unit produced. The y-intercept represents fixed costs.