A group of enterprising engineering students has developed a process for extracting combustible methane gas from cow manure​ (don't worry, the exhaust is​ odorless).With a specially adapted internal combustion​ engine, the students claim that an automobile can be propelled 15 miles per day from the​ "cow gas" produced by a single cow. Their experimental car can travel 55 miles per day for an estimated cost of ​$8 ​(this is the allocated cost of the methane process​ equipment, the cow manure is essentially​ free). Assume that there are 365 days in a year.
a. How many cows would it take to fuel 1,000,000 miles of annual driving by a fleet of​ cars? What is the annual​ cost?
b. How does your answer to Part​ (a) compare to a​ gasoline-fueled car averaging 23 miles per gallon when the cost of gasoline is ​$3.34 per​ gallon?

Let's work through the problem step by step:

### Part a: How many cows are required to fuel 1,000,000 miles of annual driving?

#### Information provided:
1. A single cow can produce enough methane to propel a car **15 miles per day**.
2. There are **365 days** in a year.
3. The car can travel **15 miles per day** from the methane produced by one cow.

#### Step 1: Calculate the annual mileage produced by a single cow.
A single cow allows a car to travel:
\[
15 \text{ miles/day} \times 365 \text{ days/year} = 5,475 \text{ miles/year}
\]
So, each cow can fuel **5,475 miles per year**.

#### Step 2: Calculate the number of cows needed for 1,000,000 miles.
To drive **1,000,000 miles per year**, the number of cows required is:
\[
\frac{1,000,000 \text{ miles}}{5,475 \text{ miles/cow}} \approx 182.71
\]
Since the number of cows must be a whole number, we round up to **183 cows**.

#### Step 3: Calculate the annual cost of running these 183 cows.
The cost for each car to travel **55 miles per day** is **$8/day**. Therefore, the daily cost per car is:
\[
\frac{8 \text{ dollars}}{55 \text{ miles}} \times 1,000,000 \text{ miles} = \frac{8 \times 1,000,000}{55} = 145,454.55 \text{ dollars/year}
\]

So, the **annual cost** for running 183 cows is approximately **$145,454.55**.

---

### Part b: Comparison with a gasoline-fueled car.

#### Information provided:
1. The gasoline-fueled car averages **23 miles per gallon**.
2. The cost of gasoline is **$3.34 per gallon**.

#### Step 1: Calculate the number of gallons needed for 1,000,000 miles.
The number of gallons of gasoline required to drive 1,000,000 miles is:
\[
\frac{1,000,000 \text{ miles}}{23 \text{ miles/gallon}} \approx 43,478.26 \text{ gallons}
\]

#### Step 2: Calculate the total cost for gasoline.
The total cost of gasoline is:
\[
43,478.26 \text{ gallons} \times 3.34 \text{ dollars/gallon} \approx 145,217.97 \text{ dollars}
\]

So, the **total cost for gasoline** would be approximately **$145,217.97**.

---

### Conclusion:
- The **annual cost** for running the cars using methane from cows is **$145,454.55**.
- The **annual cost** for running the same cars using gasoline is **$145,217.97**.
- The cost difference is minimal, with methane being slightly more expensive than gasoline by about **$236.58** annually.

Would you like more details or have any questions?  
Here are some related questions:

1. How does the environmental impact compare between methane and gasoline?
2. What would happen if the cost of gasoline increased to $4.00 per gallon?
3. How efficient is the methane combustion process compared to gasoline?
4. What other factors might affect the feasibility of using methane from cows for fueling cars?
5. Could using methane gas from cows help reduce greenhouse gas emissions?

**Tip**: Renewable energy sources, like methane from cows, can help diversify fuel options and reduce dependence on fossil fuels.

A group of enterprising engineering students has developed a process for extracting combustible methane gas from cow manure​ (don't worry, the exhaust is​ odorless). With a specially adapted internal combustion​ engine, the students claim that an automobile can be propelled 15 miles per day from the​ 'cow gas' produced by a single cow. Their experimental car can travel 55 miles per day for an estimated cost of ​$8 ​(this is the allocated cost of the methane process​ equipment, the cow manure is essentially​ free). Assume that there are 365 days in a year. a. How many cows would it take to fuel 1,000,000 miles of annual driving by a fleet of​ cars? What is the annual​ cost? b. How does your answer to Part​ (a) compare to a​ gasoline-fueled car averaging 23 miles per gallon when the cost of gasoline is ​$3.34 per​ gallon?

Discover how many cows are needed to fuel 1,000,000 miles of driving annually using methane, and compare the cost to a gasoline-powered car. Learn the cost breakdown for both fuels, including formulas for consumption rates and annual costs. Suitable for high school students learning about cost analysis and rates.

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