# An Entrepreneur's Dilemma

### The Chicken and Duck Egg Controversy

## Scenario: Farm Fresh Eggs

## G.R.A.S.S

**G**__iven__: 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

**R**__equired____:__ To conclude which animal would be the best choice for the entrepreneur's farm

**A**__nalyze____:__ 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

**S**__olve____:__ 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__**: 4 chicken Eggs Laid**

__0____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 Duck Eggs Laid**

__0__**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

## My Graph

## The Cartesian plane Below illustrates the same scenario and point of intersection as my graph above; however, it is completed using desmos graphing calculator

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

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.