Problem 96 - Algebra A Combined Function

Question

Basal metabolic rate \((B M R)\) is the number of calories per day a person needs to maintain life. A person's basal metabolic rate \(B(w)\) in calories per day can be estimated with the function \(B(w)=70 w^{3 / 4},\) where \(w\) is the person's weight in kilograms. Use this information to answer Exercises 95 and \(96 .\) Estimate the BMR for a person who weighs 90 kilograms. Round to the nearest calorie. (Note: 90 kilograms is approximately 198 pounds.)

Step-by-Step Solution

Answer

The estimated BMR is 2078 calories per day.

Step 1: Understand the Problem

We need to estimate the Basal Metabolic Rate (BMR) for a person weighing 90 kilograms using the given formula \( B(w) = 70w^{3/4} \).

Step 2: Substitute the Given Weight

Substitute \( w = 90 \) into the formula. Thus, the expression becomes \( B(90) = 70 (90)^{3/4} \).

Step 3: Calculate the Exponent

Calculate \( 90^{3/4} \). This involves taking the 4th root of 90 and then raising the result to the 3rd power. Use a calculator for precision: \( 90^{0.75} \approx 29.680 \).

Step 4: Multiply by 70

Multiply the result from the previous step by 70: \( 70 \times 29.680 \approx 2077.6 \).

Step 5: Round to Nearest Calorie

Round 2077.6 to the nearest whole number, resulting in 2078.

Key Concepts

Caloric Needs

Caloric needs refer to the number of calories a person requires daily to maintain essential physiological functions at rest, known as the Basal Metabolic Rate (BMR). The BMR is a crucial aspect of calculating caloric needs because it outlines the energy expenditure when the body is at complete rest. This includes the energy necessary for vital functions such as breathing, circulation, cell production, nutrient processing, and maintaining body temperature. Understanding one's caloric needs is important for maintaining, losing, or gaining weight. To accurately determine these needs, factors such as age, sex, weight, and activity level are considered. However, the BMR provides a foundational starting point for understanding the body's baseline energy requirements.

Weight in Kilograms

Weight in kilograms is a fundamental unit of measurement for the mass of a person or object in the metric system. In scientific and health-related contexts, kilograms are preferred over pounds because they are part of the International System of Units (SI), which is widely used worldwide. When determining the BMR, the weight in kilograms serves as a key variable in mathematical formulas, like the one used in the exercise, which is essential for calculating the energy expenditure at rest. For instance, in the formula \( B(w) = 70w^{3/4} \), \( w \) represents the weight in kilograms and directly influences the outcome, hence altering the individual's caloric needs.

Mathematical Function

A mathematical function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In the context of BMR, the function \( B(w) = 70w^{3/4} \) illustrates how a person's weight influences their basal metabolic rate. Here, \( B(w) \) is the function that provides the total calories needed at rest based on the input weight \( w \). This function is critical in understanding how changing the input (weight) can affect the output (BMR). Functions are utilized across various fields to model relationships, predict outcomes, and solve problems efficiently.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It denotes repeated multiplication of a number by itself. For instance, in \( w^{3/4} \), \( w \) is the base and \( 3/4 \) is the exponent. This means you take the base \( w \) to the power of \( 3/4 \). Breaking this down involves two steps: first, finding the fourth root of the base and then cubing the result. In practical terms, exponentiation allows for adjustments to the effects of weight on BMR, capturing non-linear relationships. It provides a more nuanced representation of how metabolic rates change with weight, allowing for more accurate caloric estimations.