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Chapter 16: Problem 26

Sleeping Metabolic Rate When people sleep, their metabolic rate is about \(2.6 \times 10^{-4} \mathrm{C} /(\mathrm{s} \cdot \mathrm{kg}) .\) How many Calories does a 75 -kg person metabolize while getting a good night's sleep of \(8.0 \mathrm{hr}\) ?

Short Answer

Expert verified
A 75 kg person metabolizes 562 Calories in 8 hours of sleep.

Step by step solution

01

Convert hours to seconds

First, we need to convert the time from hours to seconds since the metabolic rate is given per second. There are 3600 seconds in one hour. Therefore, the total sleep time in seconds is:\[8.0 \text{ hours} \times 3600 \text{ seconds/hour} = 28800 \text{ seconds}.\]
02

Calculate total metabolic rate in calories per kilogram

Next, we calculate the total metabolic rate in calories per kilogram. This is found by multiplying the metabolic rate by the time in seconds:\[2.6 \times 10^{-4} \text{ Calorie/(s kg)} \times 28800 \text{ s} = 7.488 \text{ Calorie/kg}.\]
03

Calculate total calories metabolized for 75 kg

Finally, we find out how many calories a 75 kg person metabolizes. Multiply the total calories per kilogram by the person's weight:\[7.488 \text{ Calorie/kg} \times 75 \text{ kg} = 561.6 \text{ Calorie}.\]
04

Conclusion: Round to appropriate significant figures

The calculation yields 561.6 Calories. Considering significant figures (since the given metabolic rate has two significant figures), the final answer should be:\[562 \text{ Calories}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conversion of Time Units
When converting time from hours to seconds, remember that there are 3600 seconds in every hour. This conversion is crucial in situations where measurements or rates are specified in seconds, as is the case with metabolic rate.
  • To convert hours to seconds, multiply the number of hours by 3600.
  • For example, 8 hours would be converted to seconds by calculating \(8 \times 3600\), which equals 28800 seconds.
This step ensures that units are consistent with each other, which is essential for accurate calculations.
Calories Calculation
Calculating calories involves understanding how metabolic rates work over a period. In this case, the metabolic rate is given as calories per second per kilogram.
  • First, multiply the metabolic rate \(2.6 \times 10^{-4} \, \text{Cal/s/kg}\) by the total time in seconds (28800 s).
  • This gives the total calories metabolized per kilogram: \(2.6 \times 10^{-4} \times 28800 = 7.488 \, \text{Cal/kg}\).
Understanding this process helps ensure that the energy expended through metabolic processes is accurately represented.
Significant Figures
Significant figures are critical in scientific calculations as they convey the precision of measurements. When performing calculations, it is vital to round the results to the correct number of significant figures based on the provided data.
  • For instance, in the given exercise, the metabolic rate has two significant figures due to the number \(2.6\).
  • Thus, despite the raw calculation yielding \(561.6\), the final answer is rounded to 562 calories to match the precision of the initial measurements.
This attention to detail helps maintain consistency and accuracy in findings.
Weight Calculation
When calculating total calories based on weight, it鈥檚 essential to apply the metabolic rate across the entire body mass. Once the metabolic rate per kilogram is given, as in \(7.488 \, \text{Cal/kg}\), you scale it by the person's weight.
  • Multiply the calories per kilogram by the weight in kilograms: \(7.488 \times 75 = 561.6 \, \text{Calories}\).
  • This step gives the overall energy expended during a specified time鈥攊n this case, sleep.
Understanding these conversion processes leads to more precise caloric estimations tied effectively to body weight.

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