Let's go through each part of the question step-by-step, using the data provided:
### Part (a): Checking if the Relationship is Proportional
To determine if the relationship between the amount of dry dog food and the amount of wet dog food is proportional, we need to check if the ratio of dry dog food to wet dog food is constant across all the given pairs.
#### Step 1: Calculate the ratios for each pair
1. For the first pair:
[tex]\[
\text{Ratio}_1 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{4.5 \text{ cups}}{1.5 \text{ oz}} = 3.0
\][/tex]
2. For the second pair:
[tex]\[
\text{Ratio}_2 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{6 \text{ cups}}{2 \text{ oz}} = 3.0
\][/tex]
3. For the third pair:
[tex]\[
\text{Ratio}_3 = \frac{\text{Dry dog food}}{\text{Wet dog food}} = \frac{9 \text{ cups}}{3 \text{ oz}} = 3.0
\][/tex]
#### Step 2: Compare the ratios
Since all the calculated ratios are equal (3.0, 3.0, 3.0), the ratio of dry dog food to wet dog food is constant.
#### Conclusion for Part (a):
The relationship between the amount of dry dog food and wet dog food is proportional because the ratio is constant.
### Part (b): Predicting the Amount of Dry Dog Food for 5.5 oz Wet Dog Food
Since we have established that the relationship is proportional, we can use the constant ratio to determine the amount of dry dog food needed for 5.5 oz of wet dog food.
#### Step 1: Use the known ratio
We know that:
[tex]\[
\text{Ratio} = 3.0
\][/tex]
#### Step 2: Set up the proportion
To find the amount of dry dog food needed for 5.5 oz of wet dog food, use the ratio:
[tex]\[
\frac{\text{Dry dog food}}{\text{Wet dog food}} = 3.0
\][/tex]
Let [tex]\( x \)[/tex] be the amount of dry dog food required for 5.5 oz of wet dog food. Then:
[tex]\[
\frac{x}{5.5} = 3.0
\][/tex]
#### Step 3: Solve for [tex]\( x \)[/tex]
Multiply both sides of the equation by 5.5 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3.0 \times 5.5 = 16.5
\][/tex]
#### Conclusion for Part (b):
The kennel should use 16.5 cups of dry dog food with 5.5 oz of wet dog food.
