A botanist models the rate of water transpiration of a plant species under various tempera… — SAT

A botanist models the rate of water transpiration of a plant species under various temperature conditions. The table below shows the estimated transpiration rate, $R$ , in milligrams of water per square decimeter of leaf area per hour ( $\text{mg}/(\text{dm}^2\cdot\text{h})$ ), at various ambient temperatures, $t$ , in degrees Celsius ( $^\circ\text{C}$ ).

Temperature ( $t$ )	Transpiration rate ( $R$ )
$15$	$84.5$
$20$	$108.5$
$25$	$132.5$
$30$	$156.5$

The relationship between the ambient temperature and the transpiration rate is linear. Based on the model, what is the estimated increase in the transpiration rate, in milligrams of water per square meter of leaf area per minute, for each increase of $1$ degree Celsius in the ambient temperature? (Note: $1\text{ square meter} = 100\text{ square decimeters}$ )

Answer: 8 mg/(m^2*min)

Answer

To find the estimated increase in transpiration rate per degree Celsius in the new units, we first determine the rate of change in the original units from the table. The change in temperature is

20 - 15 = 5^\circ\text{C}

, and the corresponding change in transpiration rate is

108.5 - 84.5 = 24\text{ mg}/(\text{dm}^2\cdot\text{h})

. The rate of change is

\frac{24}{5} = 4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

temperature increase. Converting this rate to square meters, we multiply by

100

because

1\text{ square meter} = 100\text{ square decimeters}

, giving

480\text{ mg}/(\text{m}^2\cdot\text{h})

. Finally, to convert to minutes, we divide by

60

because

1\text{ hour} = 60\text{ minutes}

, which yields

\frac{480}{60} = 8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

temperature increase.

Step-by-Step Solution

Find the rate of change of the transpiration rate with respect to temperature from the given data table.

Slope =

4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

The rate of change represents the increase in transpiration rate for each

1^\circ\text{C}

increase in temperature in the original units.

Convert the unit of area in the rate of change from square decimeters to square meters.

Rate =

480\text{ mg}/(\text{m}^2\cdot\text{h})

per

1^\circ\text{C}

Since

1\text{ square meter} = 100\text{ square decimeters}

, the rate per square meter is

100

times the rate per square decimeter.

Convert the unit of time in the rate of change from hours to minutes.

Rate =

8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

Since there are

60

minutes in an hour, dividing the hourly rate by

60

gives the rate per minute.

Key Concept

Interpreting the slope of a linear relationship in context and performing unit conversions.

Estimated Time:2m 30s

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