Question

question 1-13
the table below shows the global methane emissions from agriculture.

year	millions of tons of methane
1975	400
2000	900
2025	1500

which of the following statements is true?

• the average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 0.74 million tons per year
• the average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 1.55 million tons per year
• the average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 8.75 million tons per year
• the average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 17.3 million tons per year

Explanation:

To determine the average rate of change, we use the formula for the average rate of change of a function \( f(t) \) over the interval \([t_1, t_2]\):

$$ \text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} $$

Step 1: Identify \( t_1, t_2, f(t_1), f(t_2) \)

• \( t_1 = 1950 \) (start year), \( f(t_1) = 200 \) (methane emissions in 1950, in millions of tons).
• \( t_2 = 2025 \) (end year), \( f(t_2) = 1500 \) (methane emissions in 2025, in millions of tons).

Step 2: Calculate the change in time and emissions

• Change in time: \( t_2 - t_1 = 2025 - 1950 = 75 \) years.
• Change in emissions: \( f(t_2) - f(t_1) = 1500 - 200 = 1300 \) million tons.

Step 3: Compute the average rate of change

Substitute into the formula:

$$ \text{Average Rate of Change} = \frac{1300}{75} \approx 17.33 \, \text{? Wait, no—wait, let’s check the options again. Wait, maybe I misread the options. Wait, the options have values like 0.74, 1.55, 8.75, 17.3? Wait, no, let’s recalculate:} $$

Wait, \( 1500 - 200 = 1300 \), \( 2025 - 1950 = 75 \). Then \( \frac{1300}{75} \approx 17.33 \)? No, that can’t be. Wait, maybe the options have a typo, or I misread the emissions. Wait, the table: 1950: 200, 2025: 1500. Wait, 2025 - 1950 = 75 years. 1500 - 200 = 1300. 1300 / 75 ≈ 17.33. But the options: let’s check the options again. Wait, maybe the emissions are 200 (1950) and 1500 (2025). Wait, the options: one of them is “about 17.3 million tons per year”. Let’s verify:

\( \frac{1500 - 200}{2025 - 1950} = \frac{1300}{75} \approx 17.33 \), which is approximately 17.3 million tons per year.

Wait, but let’s check the options again. The last option says: “The average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 17.3 million tons per year.”

Wait, but let’s confirm the calculation:

\( 1500 - 200 = 1300 \) million tons over 75 years. \( 1300 \div 75 \approx 17.33 \), which matches the last option.

Answer:

The statement “The average rate of change in global methane emissions from agriculture between 1950 and 2025 is about 17.3 million tons per year” is true (the last option, depending on how the options are labeled—e.g., if the last option is D, then D. [Option Text]).