To derive the equation of the proportional relationship from the given data, we first need to establish the relationship between cost \( y \) and quantity \( x \).
From the provided points:
- For \( (2, 1.9) \)
- For \( (3, 3.27) \)
- For \( (4, 4.36) \)
Let's denote the proportionality constant as \( k \), so the relationship can be expressed as:
\[ y = kx \]
To find \( k \), we can use one of the points. Let's use the first point \( (2, 1.9) \):
\[ 1.9 = k \cdot 2 \]
Solving for \( k \):
\[ k = \frac{1.9}{2} = 0.95 \]
Now, let's verify if this constant works with the other points.
For the second point \( (3, 3.27) \):
\[ y = 0.95 \cdot 3 = 2.85 \text{ (which does not match 3.27)} \]
For the third point \( (4, 4.36) \):
\[ y = 0.95 \cdot 4 = 3.8 \text{ (which does not match 4.36)} \]
It appears the relationship is not perfectly linear through the origin with a single constant \( k \). However, if you are looking for a general linear equation, we might consider a linear regression, or you might determine a polynomial fit.
Since the data does not establish a clear direct proportionality, let’s approximate a relationship using linear regression or a similar method. If the data had shown a consistent pattern, this would have been straightforward.
For a linear regression fit, the solution can be determined using statistical methods, or from visual inspection and checking average rates of increase in cost per quantity.
In conclusion, the initial data does not support a simple proportional relationship through the origin, and a deeper statistical approach may be needed to derive a suitable equation based on the provided points. Please provide additional context or preferences for further analysis.