Problem 24 - Elementary Algebra

Question

New diets for cows that result in less belching can reduce greenhouse gas emissions. Find the maximum amount of methane gas that a herd of 60 average dairy cows produces by belching per year (1 year \(=365\) days). Write the answer using the word million. The average dairy cow belches out about 100 to 200 liters of methane each day. (Source: www.reuters.com, July 9, 2007)

Step-by-Step Solution

Answer

3.285 million liters per year

Step 1 - Determine the Average Methane Emission per Cow

Each cow produces between 100 and 200 liters of methane per day. To find the average daily methane production per cow, use the formula for the average: \[\text{Average} = \frac{100 + 200}{2} = 150 \text{ liters per day}\]

Step 2 - Calculate Yearly Methane Emission for One Cow

To find the annual emission per cow, multiply the daily average by the number of days in a year:\[150 \text{ liters/day} \times 365 \text{ days} = 54,750 \text{ liters/year}\]

Step 3 - Find Total Methane Emission for the Herd

There are 60 cows in the herd. To find the total methane emission per year, multiply the yearly emission per cow by the number of cows:\[54,750 \text{ liters/year} \times 60 = 3,285,000 \text{ liters/year}\]

Step 4 - Convert the Result to Millions of Liters

To express the total yearly emission in million liters, divide the result by 1,000,000:\[\frac{3,285,000}{1,000,000} = 3.285 \text{ million liters/year}\]

Key Concepts

Average Calculation

To solve real-world problems, it’s often necessary to calculate averages. Here, we need the average amount of methane a cow belches per day.
Each cow produces between 100 and 200 liters of methane daily. To find the average amount, add the lower and upper limits, then divide by two:
\[\text{Average} = \frac{100 + 200}{2} = 150 \text{ liters/day}\]
Knowing how to find averages is useful in many areas of algebra and statistics.
This helps simplify data and make it more understandable. Whether it's grades, costs, or emissions, understanding averages is key.

Unit Conversion

Converting units is crucial for comparing different quantities head-to-head. In this exercise, we need to find how much methane a cow emits in a year. We start with the daily average. Then, multiply it by the number of days in a year to find the annual emission.
\[\text{Yearly Emission/Cow} = 150 \text{ liters/day} \times 365 \text{ days/year} = 54,750 \text{ liters/year}\]
This conversion helps us understand the problem over a common period. Once you have unit conversion down, you can apply it universally, from minutes to hours, grams to kilograms, and so on.
Remember: always keep track of your units to make sure your calculations make sense!

Multiplication

Multiplication is the foundation of many algebraic calculations. Here, we use it to scale up our data. First, we find the yearly emission for one cow. Then, multiply by the total number of cows to get the total herd emission.
\[54,750 \text{ liters/year} \times 60 = 3,285,000 \text{ liters/year}\]
Finally, to express the result in millions, divide by 1,000,000:
\[\frac{3,285,000}{1,000,000} = 3.285 \text{ million liters/year}\]
By breaking down the multiplication steps, you can handle larger numbers easily. This method can be applied to many problems involving groups or batches. Next time you have multiple quantities, think multiplication!