- To check for proportionality, we calculate the ratio \(k = \frac{y}{x}\) for each data point: \(\frac{15}{6} = 2.5\), \(\frac{20}{8} = 2.5\), \(\frac{30}{12} = 2.5\), and \(\frac{45}{18} = 2.5\).
- Since the ratio is constant (\(k = 2.5\)), the relationship between snowfall and trucks is proportional.
- Using the linear equation \(y = 2.5x\), we calculate the number of trucks for \(x = 23\) inches: \(y = 2.5 \times 23 = 57.5\).
- The final answer for Part A is \(\boxed{True}\) and for Part B is \(\boxed{57.5}\).
Explanation
- Analyzing the Proportionality of the Relationship
The relationship between snowfall and the number of trucks is proportional if the ratio of the number of trucks to the amount of snowfall is constant for all given data points. We define the variables as \(x\) for the amount of snowfall in inches and \(y\) for the number of trucks. The given data points are \((6, 15)\), \((8, 20)\), \((12, 30)\), and \((18, 45)\). We check the constant of proportionality \(k = \frac{y}{x}\) for each pair:
- For the first point: \(\frac{15}{6} = 2.5\)
- For the second point: \(\frac{20}{8} = 2.5\)
- For the third point: \(\frac{30}{12} = 2.5\)
- For the fourth point: \(\frac{45}{18} = 2.5\)
Since the ratio \(\frac{y}{x}\) is consistently \(2.5\) for all points, the relationship is indeed proportional, and the constant of proportionality is \(k = 2.5\).
- Calculating the Number of Trucks for 23-inch Snowfall
Given that the relationship is proportional with a constant of proportionality \(k = 2.5\), we can express the number of trucks \(y\) as a function of the snowfall \(x\) using the formula \(y = 2.5x\). To find the number of trucks for a 23-inch snowfall, we substitute \(x = 23\) into the equation:
\(y = 2.5 \times 23 = 57.5\).
Since the city cannot send out a fraction of a truck, we interpret this result based on the model. In a mathematical context, the value is \(57.5\), which suggests that for 23 inches of snow, the model requires 57.5 units of capacity. In practical city planning, they would likely round up to 58 trucks.
Examples
Proportional relationships are essential for resource planning in many real-world scenarios. For example, if a bakery knows it takes exactly 0.5 kg of flour to bake one loaf of bread, the amount of flour needed is directly proportional to the number of loaves. If they need to bake 100 loaves, they can use the proportional relationship \(y = 0.5x\) to calculate that they need \(0.5 \times 100 = 50\) kg of flour. Just like the city planning for trucks based on snowfall, understanding these linear relationships helps in managing supplies, time, and labor efficiently.