A scientist measured the water pressure at different depths. She made a table showing her findings. The x-values represent the depth, in meters. The y-values represent the pressure in atmospheres (atm).

Question

x
y
0
0
15
3
30
6

Assuming there is a proportional relationship, which additional set of values could be included in the table?
(10, 2)
(40, 9)
(50, 38)
(100, 76)

Answer 1

To determine which additional set of values could be included in the table, we first need to analyze the relationship between depth (x) and pressure (y) based on the given data points:

1. At a depth of 0 meters, the pressure is 0 atm.
2. At a depth of 15 meters, the pressure is 3 atm.
3. At a depth of 30 meters, the pressure is 6 atm.

From these points, we can see that every 15 meters corresponds to an increase of 3 atmospheres of pressure.

This indicates a linear relationship where pressure increases proportionally with depth. We can express the relationship as follows:

• For every 15 meters of depth, the pressure increases by 3 atm, or the ratio of pressure per meter is: \[ \frac{3 \text{ atm}}{15 \text{ m}} = 0.2 \text{ atm/m} \]

Using this relationship, we can derive the following equation for pressure (y) at depth (x): \[ y = 0.2x \]

Now we'll check the potential additional values:

1. (10, 2): \[ y = 0.2 \cdot 10 = 2 \text{ atm} \quad \text{(valid)} \]

2. (40, 9): \[ y = 0.2 \cdot 40 = 8 \text{ atm} \quad \text{(not valid)} \]

3. (50, 38): \[ y = 0.2 \cdot 50 = 10 \text{ atm} \quad \text{(not valid)} \]

4. (100, 76): \[ y = 0.2 \cdot 100 = 20 \text{ atm} \quad \text{(not valid)} \]

The only additional set of values that could be included in the table, with respect to the proportional relationship between depth and pressure, is (10, 2).