C). A nutrition enthusiast is weighing the pros and cons of keeping a few chickens in his backyard, reasoning that eggs that he doesn't eat he can sell to friends and neighbors. One trip to his local hardware store later, he returns with $560 worth of lumber, paint and carpet and a weekend of fun assembling a chicken coop. He starts small, and invests another $20 in two barred rock and two Buff Orpington hens. A 50-pound sack of layer pellets costs $12 and water is essentially free. Much to his dismay, his doctor tells him his cholesterol is high, so he elects to sell all of his eggs to raise money for a popular statin medication. It takes the four hens a month to work their way through the sack of feed and during that time he collects 84 eggs, which he sells for $5 per dozen.

i)What is his break-even point in dozens? (3 marks)
ii) What is his profit the first month?
iii) How many months pass before he realizes a profit from this enterprise?


Let's break this down step by step.

### Given Information:
- Initial investment in lumber, paint, and carpet: **$560**.
- Investment in hens: **$20**.
- 50-pound sack of feed costs: **$12**.
- Number of eggs collected per month: **84 eggs**.
- Eggs sold for: **$5 per dozen**.
- Time to consume one sack of feed: **1 month**.
- Feed cost per month: **$12**.

### i) **Break-even point in dozens**

The break-even point occurs when the revenue equals the total cost (both fixed and variable costs).

#### Fixed costs:
- **Coop**: $560 (one-time cost).
- **Hens**: $20 (one-time cost).
  
Total **fixed cost**: $560 + $20 = **$580**.

#### Variable costs:
- **Feed**: $12 per month (recurring).

Revenue per dozen of eggs is **$5**.

To find the break-even point, we first calculate the revenue per month based on the number of dozens of eggs sold:
- Number of dozens per month = $ \frac{84 \text{ eggs}}{12 \text{ eggs per dozen}} = 7 \text{ dozen} $.

Revenue per month = $ 7 \text{ dozen} \times 5 \text{ dollars per dozen} = 35 \text{ dollars} $.

Now, let's calculate how many dozens he needs to sell to cover the fixed and variable costs:
- Let $ x $ be the number of dozens needed to break even.

Since each dozen earns him $5 and the cost of feed is $12 per month:
\[
5x - 12 = 580 \quad \text{(since he needs to cover fixed costs and feed cost)}
\]
\[
5x = 580 + 12 = 592
\]
\[
x = \frac{592}{5} = 118.4
\]

So, he needs to sell **118.4 dozens** to break even. Since we can't sell a fraction of a dozen, rounding up, he needs to sell **119 dozens** to break even.

### ii) **Profit in the first month**

In the first month, the fixed costs are already accounted for, but the variable costs (feed) remain.

- Revenue in the first month = $ 7 \text{ dozen} \times 5 = 35 \text{ dollars} $.
- Variable cost (feed) in the first month = $ 12 \text{ dollars} $.

So, the profit in the first month is:
\[
\text{Profit} = \text{Revenue} - \text{Variable cost} = 35 - 12 = 23 \text{ dollars}.
\]

However, since he has not yet covered the fixed costs, this $23 would go towards reducing the initial investment, not profit.

### iii) **Months to realize a profit**

To calculate the number of months needed to recoup the initial investment, we need to figure out how long it takes him to cover the **fixed costs** of $580.

Each month, after covering the feed cost, he has a net income of $23:
- Monthly net income = $ 35 - 12 = 23 \text{ dollars} $.

Now, divide the total fixed cost by the net income per month:
\[
\text{Months to break even} = \frac{580}{23} \approx 25.22 \text{ months}.
\]

Since the number of months must be a whole number, it will take him **26 months** to fully cover the fixed costs and start making a profit.

### Summary of answers:
i) **Break-even point**: 119 dozens.
ii) **Profit in the first month**: $23 (goes towards fixed costs).
iii) **Months to realize a profit**: 26 months.

---

Would you like more details on any of these calculations? 

Here are 5 related questions you might find interesting:
1. What if the feed cost increased by 10%—how would that affect the break-even point?
2. If he decided to eat some eggs, how would it impact his profit timeline?
3. How much would he need to increase the egg price to reduce the break-even period by 5 months?
4. What is the impact of purchasing more hens on his break-even point?
5. How much does each hen contribute to his monthly revenue?

**Tip:** Fixed costs only impact break-even calculations in the long run, while variable costs affect every month's profit.

A nutrition enthusiast is weighing the pros and cons of keeping a few chickens in his backyard, reasoning that eggs that he doesn't eat he can sell to friends and neighbors. One trip to his local hardware store later, he returns with $560 worth of lumber, paint and carpet and a weekend of fun assembling a chicken coop. He starts small, and invests another $20 in two barred rock and two Buff Orpington hens. A 50-pound sack of layer pellets costs $12 and water is essentially free. Much to his dismay, his doctor tells him his cholesterol is high, so he elects to sell all of his eggs to raise money for a popular statin medication. It takes the four hens a month to work their way through the sack of feed and during that time he collects 84 eggs, which he sells for $5 per dozen. i) What is his break-even point in dozens? ii) What is his profit the first month? iii) How many months pass before he realizes a profit from this enterprise?

Learn how to calculate the break-even point and monthly profit for a small backyard chicken business. This problem involves fixed and variable cost analysis, selling eggs by the dozen, and how long it takes to realize a profit. Suitable for students in Grades 9-12.

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