Question
Scientists are studying how the average size of Arctic sea ice has changed over the years. Using satellite data, they generate a scatter plot showing the relationship between the average sea ice coverage (in million square kilometers) and the number of years since 1980. A line of best fit is modeled by the equation: S(x)=7.542−0.031x where S(x) is the average sea ice coverage, and x is the number of years since 1980. (3 points) Part A: Interpret the 𝑦-intercept of the linear equation modeling the data. The model suggests that in the year 2000 the average Arctic sea ice coverage was approximately million square kilometers. Part B: Interpret the slope of the linear equation that models the data. The average Arctic sea ice coverage is decreasing by approximately million square kilometers each year. Part C: According to the model, the Arctic sea ice coverage will decrease by approximately million square kilometers over a 10 year period.
Answers
GPT-4o mini
Let's break down each part of the question regarding the average size of Arctic sea ice and the given linear equation, \( S(x) = 7.542 - 0.031x \).
### Part A: Interpret the \( y \)-intercept of the linear equation.
The \( y \)-intercept of the equation is 7.542, which occurs when \( x = 0 \) (the year 1980). This means that the model suggests that in the year 1980, the average Arctic sea ice coverage was approximately **7.542 million square kilometers**.
### Part B: Interpret the slope of the linear equation that models the data.
The slope of the linear equation is -0.031. This indicates that the average Arctic sea ice coverage is decreasing by approximately **0.031 million square kilometers** each year. This can also be expressed as **31,000 square kilometers** per year.
### Part C: According to the model, how much will the Arctic sea ice coverage decrease over a 10 year period?
To find the total decrease over a 10-year period, we can multiply the slope by the number of years:
\[
\text{Decrease over 10 years} = \text{slope} \times 10 = -0.031 \text{ million square kilometers/year} \times 10 \text{ years} = -0.31 \text{ million square kilometers}
\]
Thus, according to the model, the Arctic sea ice coverage will decrease by approximately **0.31 million square kilometers** over a 10-year period.
### Part A: Interpret the \( y \)-intercept of the linear equation.
The \( y \)-intercept of the equation is 7.542, which occurs when \( x = 0 \) (the year 1980). This means that the model suggests that in the year 1980, the average Arctic sea ice coverage was approximately **7.542 million square kilometers**.
### Part B: Interpret the slope of the linear equation that models the data.
The slope of the linear equation is -0.031. This indicates that the average Arctic sea ice coverage is decreasing by approximately **0.031 million square kilometers** each year. This can also be expressed as **31,000 square kilometers** per year.
### Part C: According to the model, how much will the Arctic sea ice coverage decrease over a 10 year period?
To find the total decrease over a 10-year period, we can multiply the slope by the number of years:
\[
\text{Decrease over 10 years} = \text{slope} \times 10 = -0.031 \text{ million square kilometers/year} \times 10 \text{ years} = -0.31 \text{ million square kilometers}
\]
Thus, according to the model, the Arctic sea ice coverage will decrease by approximately **0.31 million square kilometers** over a 10-year period.