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Difficulty: MediumInterpreting Linear Relationships in Context

A commercial hydroponic farm monitors the volume of nutrient solution in a reservoir. The table below shows the volume of solution remaining, VV, in liters, after tt hours of operation.

Time (tt, hours)Volume (VV, liters)
0450
10432
20414
30396

The relationship between the time and the volume of remaining solution can be modeled by a linear equation. Which of the following is the best interpretation of the slope of the graph of this equation in the tVtV-plane?

  1. The nutrient solution is being depleted at a rate of 1.8 liters per hour.Answer
  2. B
    The initial volume of the nutrient solution in the reservoir is 450 liters.
  3. C
    The reservoir will be completely empty after 250 hours.
  4. D
    The volume of the nutrient solution decreases by 1 liter every 1.8 hours.

Answer

The nutrient solution is being depleted at a rate of 1.8 liters per hour.
The slope of a linear function represents the constant rate of change of the dependent variable with respect to the independent variable. By selecting two data points from the table, such as (0,450)(0, 450) and (10,432)(10, 432), the slope is calculated as 432−45010−0=−1.8\frac{432 - 450}{10 - 0} = -1.8. Since the volume VV is measured in liters and the time tt is in hours, this slope represents a decrease of 1.81.8 liters of nutrient solution per hour of operation. Therefore, the nutrient solution is being depleted at a rate of 1.81.8 liters per hour.

Step-by-Step Solution

1
Calculate the slope using two points from the table.
Using the points (0,450)(0, 450) and (10,432)(10, 432), the slope is 432−45010−0=−1810=−1.8\frac{432 - 450}{10 - 0} = \frac{-18}{10} = -1.8.
The slope of a linear relationship is given by the change in the dependent variable (Volume) divided by the change in the independent variable (Time).
2
Interpret the unit of the slope in context.
The unit of the slope is liters per hour (L/h\text{L/h}), representing a change of −1.8-1.8 liters for every hour of operation.
Determining the units of the rate helps verify what physical quantity the slope represents.
3
Relate the negative sign of the slope to the context of depletion.
A negative slope of −1.8-1.8 means the volume of nutrient solution decreases by 1.81.8 liters each hour, which translates to a depletion rate of 1.81.8 liters per hour.
Connecting the mathematical sign to the real-world action clarifies whether the quantity is increasing or decreasing.

Key Concept

Interpreting the slope of a linear equation as a constant rate of change in a real-world context.
Estimated Time:1m 30s
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Topics

Interpreting Linear Relationships in Context

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