Let's carefully analyze the data and determine which additional set of values could be included in the table, given that there's a proportional relationship between depth and pressure.
### Step-by-Step Solution
1. Understanding the Proportional Relationship:
- In a proportional relationship, the ratio [tex]\( \frac{y}{x} \)[/tex] (pressure to depth) is constant.
2. Given Data:
- Depth (x): 0, 15, 30 meters
- Pressure (y): 0, 3, 6 atmospheres
3. Finding the Constant of Proportionality (k):
- Choose any known pair of depth and pressure to calculate [tex]\( k \)[/tex].
- Using the pair (15, 3):
[tex]\[
k = \frac{y}{x} = \frac{3}{15} = 0.2
\][/tex]
4. Applying the Proportional Relationship:
- Using the constant [tex]\( k = 0.2 \)[/tex], we can write the relationship as:
[tex]\[
y = 0.2 \cdot x
\][/tex]
5. Testing the Given Choices:
- Check which of the given choices fit the equation [tex]\( y = 0.2 \cdot x \)[/tex]:
- Choice (10, 2):
[tex]\[
y = 0.2 \cdot 10 = 2
\][/tex]
This fits because [tex]\( y = 2 \)[/tex].
- Choice (40, 9):
[tex]\[
y = 0.2 \cdot 40 = 8
\][/tex]
This does not fit because [tex]\( y \neq 9 \)[/tex].
- Choice (50, 38):
[tex]\[
y = 0.2 \cdot 50 = 10
\][/tex]
This does not fit because [tex]\( y \neq 38 \)[/tex].
- Choice (100, 76):
[tex]\[
y = 0.2 \cdot 100 = 20
\][/tex]
This does not fit because [tex]\( y \neq 76 \)[/tex].
6. Conclusion:
- The only set of values in the given choices that maintains the proportional relationship is [tex]\( (10, 2) \)[/tex].
Therefore, the set of values that could be included in the table while maintaining the proportional relationship is [tex]\( (10, 2) \)[/tex].
