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Difficulty: MediumInterpreting Linear Relationships in Context

A botanist is studying the transpiration rate of a certain plant species under controlled conditions. The transpiration rate TT, in grams of water vapor per square meter of leaf area per hour (g/m2/h\text{g/m}^2/\text{h}), is modeled by a linear function of the relative humidity HH, expressed as a decimal where 0.20≤H≤0.800.20 \leq H \leq 0.80. The model is represented by the equation T=−4.8H+CT = -4.8H + C, where CC is a constant. Based on this model, if the relative humidity increases by 0.150.15, what is the corresponding decrease in the transpiration rate, in g/m2/h\text{g/m}^2/\text{h}?

Answer: 0.72 g/m^2/h

Answer

The correct answer is 0.72.
In the linear equation T=−4.8H+CT = -4.8H + C, the coefficient of HH (which is −4.8-4.8) represents the slope of the line. The slope indicates that for every increase of 1.01.0 in the relative humidity HH, the transpiration rate TT decreases by 4.84.8 grams of water vapor per square meter of leaf area per hour. To find the decrease in transpiration rate corresponding to an increase of 0.150.15 in relative humidity, multiply the rate of change by the change in relative humidity: 4.8×0.15=0.724.8 \times 0.15 = 0.72. Thus, the transpiration rate decreases by 0.720.72 grams of water vapor per square meter of leaf area per hour.

Step-by-Step Solution

1
Identify the slope of the linear model.
The slope is −4.8-4.8.
The coefficient of HH in the linear equation T=−4.8H+CT = -4.8H + C represents the rate of change of the transpiration rate with respect to the relative humidity.
2
Calculate the change in the transpiration rate (ΔT\Delta T) for an increase of 0.150.15 in the relative humidity (ΔH=0.15\Delta H = 0.15).
ΔT=−4.8×0.15=−0.72\Delta T = -4.8 \times 0.15 = -0.72
The change in the dependent variable is equal to the slope multiplied by the change in the independent variable.
3
Determine the magnitude of the decrease.
The decrease in the transpiration rate is 0.720.72.
The negative sign in the change ΔT=−0.72\Delta T = -0.72 represents a decrease, so the amount of decrease is 0.720.72.

Key Concept

Interpreting the slope of a linear relationship in context as the rate of change of the dependent variable with respect to the independent variable.
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Interpreting Linear Relationships in Context

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