The function C=500+0.5xx−50

Question

gives the average cost of producing x widgets, where the startup cost is $500. Once the widgets are developed, they are manufactured for $0.50 each. The first 50 widgets will not be sold but will be produced as free samples.

What do the key features of the function reveal?

(1 point)
Responses

The range is C≤0
, so the average cost only has negative values.
The range is cap c is less than or equal to 0 , so the average cost only has negative values.

The function has a horizontal asymptote at C=0.5
, so as the number of widgets produced increases, the average cost per widget produced approaches $0.50.
The function has a horizontal asymptote at cap c is equal to 0 point 5 , so as the number of widgets produced increases, the average cost per widget produced approaches $0.50.

The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate.
The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate.

The function is a decreasing function, so the average cost is decreasing at a constant rate.

Answer 1

To analyze the function \( C = 500 + 0.5x - 50 \), first, let's simplify it:

\[ C = 500 - 50 + 0.5x = 450 + 0.5x. \]

This function represents the average cost of producing \( x \) widgets.

Key Features:

1. Startup Cost: The constant \( 450 \) indicates that there is a fixed cost of production, which accounts for the startup costs. In this scenario, the initial \( $500 \) includes both fixed costs and the effective offset of the free samples given (the first 50 widgets).

2. Variable Production Cost: The term \( 0.5x \) indicates that for every additional widget produced, it costs an additional \( $0.50 \).

3. Horizontal Asymptote: As the number of widgets produced increases, the average cost \( C \) will approach \( 0.5 \) for large values of \( x \), because the fixed cost becomes less significant relative to the total cost as \( x \) increases. However, it is not correct to say that there is a horizontal asymptote at \( C = 0.5 \); instead, the average cost consists of a variable term \( 0.5x \) that continues to rise indefinitely with \( x \).

4. Increasing Function: The function \( C(x) = 450 + 0.5x \) is indeed an increasing function. The rate of increase is not constant since it is linear, which represents a steady increase in average costs as more widgets are produced.

Conclusion:

Among the provided responses, the most accurate points would be:

• The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate (this reflects the continuous increase due to the term \( 0.5x \) which keeps adding to \( C \)).

• The statement about the range being \( C \leq 0 \) is incorrect, as the costs cannot be zero or negative; they are positive values over practical production.

• While the average cost does approach the variable cost per unit (which is $0.50), it does not have a horizontal asymptote at that value since the fixed costs continue to affect the overall value in the function.

Therefore, the best fitting statement is:
"The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate."