Certainly! Let's solve this step-by-step:
### Question (a): Is the relationship between the amount of dry dog food and the amount of wet dog food proportional?
To determine if the relationship is proportional, we must check if the ratio between dry dog food and wet dog food is constant across the first three rows of data.
Step 1: Calculate the Ratio for Each Row
1. For the first row (4.6 cups of dry dog food and 1.5 oz of wet dog food):
[tex]\[
\text{Ratio}_1 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{4.6}{1.5} \approx 3.0667
\][/tex]
2. For the second row (6 cups of dry dog food and 2 oz of wet dog food):
[tex]\[
\text{Ratio}_2 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{6}{2} = 3.0
\][/tex]
3. For the third row (9 cups of dry dog food and 3 oz of wet dog food):
[tex]\[
\text{Ratio}_3 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{9}{3} = 3.0
\][/tex]
Step 2: Compare the Ratios
The calculated ratios are as follows:
- [tex]\(\text{Ratio}_1 \approx 3.0667\)[/tex]
- [tex]\(\text{Ratio}_2 = 3.0\)[/tex]
- [tex]\(\text{Ratio}_3 = 3.0\)[/tex]
Since [tex]\(\text{Ratio}_1 \ne \text{Ratio}_2\)[/tex], the ratios are not consistent, thus the relationship between the amount of dry dog food and wet dog food is not proportional.
### Question (b): How much dry dog food is used with 5.5 oz of wet dog food?
Since we've determined that the relationship is not proportional, we cannot use a direct proportionality to determine the amount of dry dog food for 5.5 oz of wet dog food. Therefore, it isn't straightforward to answer this using a constant ratio. Hence, we can consider the amount of dry dog food used with 5.5 oz of wet dog food to be undefined under the assumption of proportionality.
To directly address part (b):
Given that the relationship is confirmed not to be proportional:
- The amount of dry dog food for 5.5 oz of wet dog food cannot be directly calculated due to the inconsistency in the ratios.
- Explicitly, the desired dry dog food amount is marked as None (or undefined).
This concludes the detailed step-by-step solution for the given problem.
