- We analyze the relationship between snowfall (\(y\)) and time (\(x\)) by checking if the ratio \(\frac{y}{x}\) is constant.
- We calculate the ratios for each data point: \(\frac{10}{2}=5\), \(\frac{14}{6} \approx 2.33\), \(\frac{16}{8}=2\), \(\frac{5}{2.5}=2\), and \(\frac{7}{7}=1\).
- Since the ratios are inconsistent, we confirm that \(y\) is not proportional to \(x\).
- Final Answer: \(\boxed{\text{False}}\)
Explanation
- Understanding Proportionality
To determine if the snowfall \(y\) is proportional to time \(x\), we need to check if the ratio \(\frac{y}{x}\) is constant for all provided data points. A relationship is proportional if and only if \(y = k \cdot x\) for some non-zero constant \(k\), which implies that \(\frac{y}{x} = k\) for all pairs \((x, y)\). Let us examine the data points \((2, 10), (6, 14), (8, 16), (2.5, 5), (7, 7)\) given in the table.
- Evaluating the Data Pairs
We calculate the ratio \(\frac{y}{x}\) for each pair:
- For \((2, 10)\): \(\frac{10}{2} = 5\)
- For \((6, 14)\): \(\frac{14}{6} = \frac{7}{3} \approx 2.33\)
- For \((8, 16)\): \(\frac{16}{8} = 2\)
- For \((2.5, 5)\): \(\frac{5}{2.5} = 2\)
- For \((7, 7)\): \(\frac{7}{7} = 1\)
Since the ratios \(\frac{y}{x}\) are not equal (e.g., \(5
eq 2.33
eq 2
eq 1\)), the relationship between the snowfall \(y\) and the time \(x\) is not proportional.
- Conclusion and Graphical Representation Analysis
If we were to graph these points \((x, y)\) on a Cartesian plane, they would not lie on a single straight line passing through the origin \((0, 0)\). For a relationship to be proportional, the points must form a straight line that passes through the origin. Since our calculated ratios are different, the points do not align with a single linear path from the origin. Therefore, we conclude that the snowfall is not proportional to the time elapsed. The answer is False.
Examples
Understanding whether relationships are proportional is essential in everyday life. For instance, if you are grocery shopping and buying apples at a fixed price per pound, the total cost is proportional to the weight of the apples. If the price per pound were to change as you bought more apples (like a bulk discount), the relationship would no longer be proportional. Similarly, in this storm scenario, if the snow fell at a constant rate, we would be able to predict the future snowfall accurately, but because the rate of snowfall changes over time, it is not a proportional relationship.