The following formula estimates an average person's lung capacity V (in liters, where 1 L = 10³ cm³): V = 4.1H ― 0.018A ―2.7, where H and A are the person's height (in meters) and age (in years), respectively. In this formula, what are the units of the numbers 4.1, 0.018, and 2.7?

검증된 단계별 안내

Step 1: Begin by analyzing the formula provided: $ V = 4.1H - 0.018A - 2.7 $. Here, $ V $ represents lung capacity in liters (L), $ H $ is height in meters (m), and $ A $ is age in years (yr). The goal is to determine the units of the constants 4.1, 0.018, and 2.7.

Step 2: Focus on the first term, $ 4.1H $. Since $ H $ has units of meters (m) and the result $ V $ must have units of liters (L), the constant 4.1 must have units of $ \text{L/m} $ to ensure the product $ 4.1H $ has units of liters.

Step 3: Next, consider the second term, $ 0.018A $. Here, $ A $ has units of years (yr), and the result must again have units of liters (L). Therefore, the constant 0.018 must have units of $ \text{L/yr} $ to ensure the product $ 0.018A $ has units of liters.

Step 4: Finally, examine the constant $ -2.7 $. Since this is a standalone term being subtracted, it must already have the same units as $ V $, which is liters (L). Thus, the constant $ -2.7 $ has units of liters.

Step 5: Summarize the findings: The units of 4.1 are $ \text{L/m} $, the units of 0.018 are $ \text{L/yr} $, and the units of 2.7 are $ \text{L} $. This ensures dimensional consistency throughout the formula.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Dimensional Analysis

Dimensional analysis is a method used in physics and engineering to check the consistency of equations by ensuring that both sides of an equation have the same dimensions or units. It helps in understanding how different physical quantities relate to each other and is crucial for determining the units of constants in formulas.

Units of Measurement

Units of measurement are standard quantities used to express physical properties. In the context of the given formula, understanding the units for height (meters) and age (years) is essential to deduce the units of the constants 4.1, 0.018, and 2.7, which must be consistent with the output unit of lung capacity (liters).

Linear Relationships

Linear relationships describe how one variable changes in relation to another in a straight-line manner. In the formula provided, the constants represent coefficients that define how height and age influence lung capacity, indicating that lung capacity increases linearly with height and decreases slightly with age.

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