Problem 113 - Intermediate Algebra

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 Estimate the BMR for a person who weighs 60 kilograms. Round to the nearest calorie. (Note: 60 kilograms is approximately 132 pounds.)

Step-by-Step Solution

Answer

The estimated BMR for a 60 kg person is 2761 calories.

Step 1: Understand the Problem

We are given a function \(B(w) = 70 w^{3/4}\) that estimates the basal metabolic rate (BMR) for a person based on their weight \(w\) in kilograms. We need to estimate the BMR for a person weighing 60 kilograms.

Step 2: Plug in the Weight Value

Substitute \(w = 60\) kilograms into the function \(B(w) = 70 w^{3/4}\). This gives us the equation: \(B(60) = 70 \times 60^{3/4}\).

Step 3: Solve the Exponent

Calculate \(60^{3/4}\). This requires finding the fourth root of 60 raised to the third power. First, calculate 60 cubed: \(60^3 = 216000\). Next, find the fourth root of 216000: \( (216000)^{1/4}\), which approximately equals 39.44.

Step 4: Multiply by 70

Take the approximate result from the previous step, 39.44, and multiply by 70: \(70 \times 39.44 = 2760.8\).

Step 5: Round the Result

The problem asks for the BMR rounded to the nearest whole calorie. Rounding 2760.8 gives 2761 calories.

Key Concepts

Calorie Calculation

Calorie calculation is essential for estimating a person's energy needs, particularly when understanding basal metabolic rate (BMR). BMR represents the number of calories your body needs to perform basic life-sustaining functions while at rest. This includes activities such as breathing, circulation, and cell production.
To estimate BMR, we use mathematical formulas like the one given in this exercise: \( B(w) = 70 w^{3/4} \). Here, \( w \) stands for the person's weight in kilograms. This formula makes it easier to approximate the energy requirements based on weight.
When calculating calories for BMR:

• Identify a person's weight in kilograms.
• Substitute this weight into the formula.
• Solve the resulting expression to find the BMR.

Understanding how to calculate calories using such formulas helps in planning diets and maintaining healthy lifestyles.

Weight Conversion

Weight conversion is the process of changing weight measurements from one unit to another, such as from kilograms to pounds or vice versa. Knowing how to convert weight is particularly useful in global contexts, where different regions use different measurement systems.
For this exercise, the given weight was 60 kilograms. To convert kilograms to pounds, you use the conversion factor:

• 1 kilogram is approximately equal to 2.20462 pounds.

Using this factor, you can convert 60 kilograms to pounds:

• 60 kilograms × 2.20462 = 132.2772 pounds.

Thus, 60 kilograms is approximately 132 pounds, as stated in the exercise. Weight conversion helps in understanding and using weight information more effectively in both day-to-day activities and scientific inquiries.

Exponent Arithmetic

Exponent arithmetic involves performing mathematical operations on numbers with exponents. It plays a vital role in many mathematical processes, including those for calculating basal metabolic rate. In this exercise, the formula \( 70 w^{3/4} \) requires you to work with exponents.
To solve \( 60^{3/4} \), you break it into simpler parts:

• First, calculate \( 60^3 \), which means multiplying 60 by itself three times: \( 60 \times 60 \times 60 = 216000 \).
• Then, find the fourth root of 216000: This can be estimated using a calculator or numerical methods to be about 39.44.

Understanding exponent arithmetic is crucial because it allows you to simplify and solve complex mathematical problems methodically. Practice with such calculations strengthens your overall algebraic skills.

Intermediate Algebra

Intermediate algebra involves exploring various algebraic concepts and skills crucial for solving real-world problems like calculating the basal metabolic rate. In this scenario, the exercise requires you to understand and apply a given formula, which is often a central task in intermediate algebra.
This process includes:

• Evaluating expressions: Substitute known values for variables.
• Solving equations: Apply arithmetic operations to find results.
• Rounding: Determine results to a specified degree of accuracy, often to the nearest whole number.

By mastering these intermediate skills, you can tackle a wide range of mathematical tasks, aiding in both academic study and practical applications. For estimating BMR, we used these skills to substitute, compute, multiply, and round numbers to arrive at a conclusive calorie count for a person's resting energy needs.