An Entrepreneur's Dilemma

The Chicken and Duck Egg Controversy

Scenario: Farm Fresh Eggs

A young entrepreneur is looking to start a farm that sells fresh eggs. He is debating whether to use chickens or ducks as his main source of fresh eggs. To decide which bird he should choose, he observes how many eggs each animal lays for over a duration of 6 weeks. Before deciding to observe this comparison, the chicken had already laid a total of 4 eggs; yet, the duck had laid none. Determine which animal would be the best choice for this young and determined entrepreneur

G.R.A.S.S

Given: The entrepreneur is observing the ducks and chickens for a duration of 6 weeks.

Before observing this comparison, the chicken had already laid 4 eggs; the duck

had laid none

Required: To conclude which animal would be the best choice for the entrepreneur's farm

Analyze: Create two formulas that would be most accurate for the specified scenario

(Let E represent the total chicken eggs laid; Let W represent the total number of weeks)

Formula for Chickens: E= W + 4

Formula for Ducks: E= 2W

Solve: Illustrate a graph to find both the intersection point, and which animal will evidently lay the most eggs after a duration of 6 weeks

State: As illustrated in the graph, after a duration of 6 weeks, the duck will lay a total of

12 eggs, while the chicken will only lay 10. Thus, the duck will evidently become

the best animal choice for the young entrepreneur's farm. Also, the point of intersection on the graph states that after a total time period of 4 weeks, both the chicken and the duck will have laid a total of 8 eggs.

A Comparison Table Between the Number of Chicken Eggs Laid, and the Total Number of Weeks Elapsed

Week 0: 4 chicken Eggs Laid

Week 1: 5 Chicken Eggs Laid

Week 2: 6 Chicken Eggs Laid

Week 3: 7 Chicken Eggs Laid

Week 4: 8 Chicken Eggs Laid

Week 5: 9 Chicken Eggs Laid

Week 6: 10 Chicken Eggs Laid

A Comparison Table Between the Number of Duck Eggs Laid, and the Total Number of Weeks Elapsed

Week 0: 0 Duck Eggs Laid

Week 1: 2 Duck Eggs Laid

Week 2: 4 Duck Eggs Laid

Week 3: 6 Duck Eggs Laid

Week 4: 8 Duck Eggs Laid

Week 5: 10 Duck Eggs Laid

Week 6: 12 Duck Eggs Laid

Identification and Classification of The Different Variables Used In This Scenario

Variables:


  • Let E represent the total number of eggs laid
  • Let W represent the total number of weeks elapsed

Equations in Slope/Y-Intercept Format

Formula For Chickens: E=W + 4

Formula For Ducks: E=2W

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My Graph

As analysed on my graph below, the chicken consists of a partial variation because it has a y-intercept of 4. Therefore, its equation in y=mx + b format is E= W + 4, where W represents the number of weeks, E represents the total number of eggs laid, and 4 represents the number of eggs laid before the observation had even started. The duck is a direct variation because it has no y-intercept. Its equation is E = 2W, where E represents the total number of eggs laid, and 2W represents that each week 2 more eggs are laid. Evidently the duck laid more eggs and was therefore the best animal choice for the entrepreneur. After a duration of 6 weeks, it laid 12 eggs while the chicken had only laid 10. The point of intersection on the graph illustrated that after a 4 week time period, both the chicken and the duck laid the same number of eggs (8 eggs).
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The Cartesian plane Below illustrates the same scenario and point of intersection as my graph above; however, it is completed using desmos graphing calculator

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A Mathematical solution Representing the Intersection point of the two Lines, as well as the Significance of the Solution

The equation for chickens is E=W + 4, and the equation for ducks is E=2W. Therefore, to solve for the intersection point, we can solve for the linear systems:


Chicken: E = W + 4 and Duck: E=2W

W + 4 = 2W

4 = 2W - W

W = 4

Therefore: (4, 0)


Then we can use this coordinate and substitute W with 4 to find the value of E. We can choose any formula to do this on because whether the formula for chickens, or ducks is chosen, they will both equal the same number. Therefore, i chose to substitute W with 4 in the chicken's formula:


E = W + 4

E = 4 + 4

E= 8

Therefore: (0, 8)


This means that the final answer is (4,8). This break-even point represents that after a duration of 4 weeks, both the duck and the chicken will lay a total of 8 eggs. We can also be assured that this answer is right by referring to the intersection point on the graph above.


Before 4 weeks (the break-even point), the chicken would have been the best option because it laid more eggs than the duck . This was seen on the graph because the chicken had a y-intercept of 4, while the duck had laid none. However, after the break-even point, the duck out-won its competition because it laid more eggs per week than the chicken. Through this analysis, the young entrepreneur is assured that the duck would be the best choice for him because it would speedily lay eggs and always keep the supply high so that there is more demand and customers for his business.

A Summary Statement

In conclusion, the point-of-intersection stated that both the chicken and the duck would lay 8 eggs in 4 weeks. However, since the entrepreneur wanted to start a business that sold fresh eggs to customers on a regular basis, it would be most appropriate to choose the animal that would most consistently and speedily lay eggs. That way the supply is always high enough to suffice the demand. Therefore, through his observation, the entrepreneur was able to analyse that the duck laid eggs at a faster rate than the chicken, and therefore would be the best animal choice for a long-term basis.
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scientific reasoning: why do ducks lay more eggs than chickens ?

It is common knowledge that ducks lay more eggs than chickens; however, the reasons behind it are not usually inquired about. In general, ducks are healthier than chickens. They are less likely to contract diseases and are less prone to mites, pesticides, and insecticides. They also have stronger immune systems and are very cold-hardy. This means that they have an extra layer of fat and waterproofing on their wings to protect their bodies from such elements that may be harmful to them. Additionally, since the male ducks leave the nest after the baby duck has been hatched, they can easily make the decision of mating with other female ducks. It is because of this behavior that ducks can easily reproduce multiple eggs over a duration of simply 6 weeks. For these reasons, ducks are said to reproduce more than chickens. I based my linear system flyer off of this theme because my grandparents own a chicken/duck farm in India. They commonly talk about how interesting it is to observe the ducks and chickens and encourage me to take an interest in it as well; therefore, i decided to base my flyer off of that topic.
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