attached

Caroline is a distributor of unique personalized PEZ dispensers. The most popular dispensers sold are team logo and baby announcements. Caroline finds that the average cost,

400 + 3x

A, of producing x number of personalized PEZ dispensers can be modeled by A(x) =

=

Little Facts: In 1927 Eduard Haas III invented the PEZ candy in Vienna, Austria, originally as a breath mint. The name PEZ is taken from the first, middle, and last letter of the

German word pfefferminz, which means peppermint. In the mid-1950s fruity flavors, along with character dispensers, were introduced to the US market, targeting children;

Popeye was the first licensed character. +

(a) Use the given function to determine the average cost of producing 60 dispensers. Round to two decimal places.

$

(b) If the average cost is $103, how many dispensers were produced?

dispensers

(c) Find the vertical asymptotes, if any. (Enter your answers as a comma-separated list. If no vertical asymptote exists, enter NONE.)

X =

Find the horizontal asymptote, if any. (If no horizontal asymptote exists, enter NONE.)

y =

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Caroline is a distributor of unique personalized PEZ dispensers that are an excellent choice for team logos and baby announcements. The cost of producing the PEZ dispensers plays a vital role in determining their selling price, and Caroline has modeled the average cost using a specific function.

Description:

Caroline has a vast collection of personalized PEZ dispensers that are extremely popular among customers, with team logo and baby announcement dispensers being the most sought-after. To determine the average cost of producing a particular number of dispensers, Caroline has developed a function that takes into account the number of dispensers produced. The function, A(x) = 400 + 3x, can be used to determine the average cost of producing a given number of PEZ dispensers, where ‘x’ is the number of dispensers produced. In this context, this article will explore the average cost of producing a set number of dispensers and determine the vertical and horizontal asymptotes of the given function.

Objectives:

1. To understand the model used to calculate the average cost of producing personalized PEZ dispensers.

2. To be able to determine the average cost of producing a specific number of PEZ dispensers using the given function.

3. To identify the vertical and horizontal asymptotes of the given function.

Learning Outcomes:

1. Students will be able to explain the significance of the average cost model used for personalized PEZ dispensers.

2. Students will be able to calculate the average cost of producing a given number of personalized PEZ dispensers using the given function.

3. Students will be able to identify the vertical asymptotes of the given function and explain the importance of their existence.

4. Students will be able to identify the horizontal asymptote of the given function, if it exists, and explain its significance in the context of personalized PEZ dispenser production.

Heading: Calculating Average Cost and Asymptotes for Personalized PEZ Dispensers

(a) Determining the Average Cost of Producing 60 PEZ Dispensers

To determine the average cost of producing 60 PEZ dispensers, we use the given function: A(x) = 400 + 3x. Substituting x = 60, we get: A(60) = 400 + 3(60) = 400 + 180 = 580. Therefore, the average cost of producing 60 PEZ dispensers is $580, rounded to two decimal places.

(b) Finding the Number of PEZ Dispensers Produced for a Given Average Cost

If the average cost is $103, we can use the given function to determine the number of dispensers produced:

$103 = 400 + 3x

3x = $103 – $400

3x = -$297

x = -$99

Since it is not possible to produce a negative number of PEZ dispensers, we can conclude that there are no dispensers produced at an average cost of $103.

(c) Identifying Vertical and Horizontal Asymptotes

To find the vertical asymptotes of the function A(x), we need to set the denominator of the function equal to zero and solve for x:

400 + 3x = 0

3x = -400

x = -400/3

Therefore, the vertical asymptote is x = -400/3.

To find the horizontal asymptote, we take the limit of A(x) as x approaches infinity:

lim A(x) = lim (400 + 3x)/x

x→ ∞ x→ ∞

Since the degree of the numerator and denominator are equal (both are 1), the horizontal asymptote is y = 3. Therefore, there is a horizontal asymptote at y = 3, but no vertical asymptote exists.

Solution 1:

(a) To determine the average cost of producing 60 dispensers using the given function, we just need to substitute x = 60 into the function A(x) = 400 + 3x. Thus, A(60) = 400 + 3(60) = $580. Therefore, the average cost of producing 60 dispensers is $580.

(b) If the average cost is $103, we can solve the given function 103 = 400 + 3x for x to find how many dispensers were produced. Subtracting 400 from both sides and dividing by 3, we get x = 33. Therefore, 33 dispensers were produced with an average cost of $103.

(c) To find the vertical asymptotes, we need to set the denominator of the given function, 1 – x/20, equal to zero and solve for x. Thus, we get 1 – x/20 = 0, which implies x = 20. Therefore, the vertical asymptote is x = 20.

(d) To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the given function. Since the degree of the numerator is less than the degree of the denominator (which is 1), the horizontal asymptote is y = 0.

Solution 2:

(a) To determine the average cost of producing 60 dispensers using the given function, we just need to substitute x = 60 into the function A(x) = 400 + 3x. Thus, A(60) = 400 + 3(60) = $580. Therefore, the average cost of producing 60 dispensers is $580.

(b) If the average cost is $103, we can solve the given function 103 = 400 + 3x for x to find how many dispensers were produced. Subtracting 400 from both sides and dividing by 3, we get x = 33. Therefore, 33 dispensers were produced with an average cost of $103.

(c) The given function A(x) has a division by (1-x/20) in its expression, which can be equal to zero only when x = 20. Hence, the vertical asymptote of the function is x = 20.

(d) As x approaches infinity or negative infinity, the term involving x in the given function A(x) becomes dominant, while the constant term 400 becomes insignificant. Hence, the horizontal asymptote of the function is y = 3x.

Suggested Resources/Books:

1. “Pez: The Collectors Guide” by Shawn Peterson

2. “The Ultimate Guide to Vintage Transformers Action Figures” by Mark Bellomo

3. “The Big Book of Pez: Identification and Price Guide” by Shawn Peterson

4. “The Comprehensive Guide to Collecting PEZ Dispensers: A Collector’s Price Guide” by Michael Edwards

Similar Asked Questions:

1. What is the history of PEZ dispensers?

2. How do you determine the cost of producing PEZ dispensers?

3. What are some popular types of personalized PEZ dispensers?

4. How do you collect and store PEZ dispensers?

5. What is the market for PEZ dispenser collectors like?

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