Problem 95 - Precalculus: Functions and Graphs

Question

Calorie reguirements The basal energy requirement for an individual indicates the minimum number of calories necessary to maintain essential life-sustaining processes such as circulation, regulation of body temperature, and respiration. Given a person's sex, weight \(w\) (in kilograms), height \(h\) (in centimeters), and age \(y\) (in years), we can estimate the basal energy requirement in calories using the following formulas, where \(C_{f}\) and \(C_{m}\) are the calories necessary for females and males, respectively: $$\begin{array}{l}C_{f}=66.5+13.8 w+5 h-6.8 y \\\C_{m}=655+9.6 w+1.9 h-4.7 y\end{array}$$ (a) Determine the basal energy requirements first for a 25 -year-old female weighing 59 kilograms who is 163 centimeters tall and then for a 55 -year-old male weighing 75 kilograms who is 178 centimeters tall. Discuss why, in both formulas, the coefficient for \(y\) is negative but the other coefficients are positive.

Step-by-Step Solution

Answer

The female needs 1525.7 calories; the male needs 1617.5 calories. Age coefficient is negative because metabolic rate decreases with age.

Step 1: Calculate Female's Basal Energy Requirement

Given: A 25-year-old female weighing 59 kg and 163 cm tall. Use the female formula: \ \( C_{f} = 655 + 9.6w + 1.9h - 4.7y \) \ Substitute the values into the formula: \ \( C_{f} = 66.5 + 13.8(59) + 5(163) - 6.8(25) \) \ Calculate each part: \ \( 66.5 + 814.2 + 815 - 170 = 1525.7 \) \ Hence, the basal energy requirement for the female is 1525.7 calories.

Step 2: Calculate Male's Basal Energy Requirement

Given: A 55-year-old male weighing 75 kg and 178 cm tall. Use the male formula: \ \( C_{m} = 66.5 + 13.8w + 5h - 6.8y \) \ Substitute the values into the formula: \ \( C_{m} = 66.5 + 13.8(75) + 5(178) - 6.8(55) \) \ Calculate each part: \ \( 66.5 + 1035 + 890 - 374 = 1617.5 \) \ Hence, the basal energy requirement for the male is 1617.5 calories.

Step 3: Discuss the Coefficients for Age (y)

In both formulas, the coefficient for \(y\) is negative because as age increases, the basal metabolic rate typically decreases due to a decrease in lean body mass and a decrease in overall metabolic activity. The other coefficients for weight and height are positive because generally, higher weight and greater height necessitate more calories to sustain essential body processes.

Key Concepts

Calories

Calories are the units that measure the energy necessary for the body to function correctly. They are vital for life-sustaining processes like breathing, circulation, and regulating body temperature. The term represented in equations as \( C \), which stands for the calorie requirement, reflects how many calories are needed daily.

When calculating calorie needs, several factors are considered:

• Sex: Different formulas account for biological differences between males and females.
• Weight: Heavier individuals generally need more energy.
• Height: Taller people have more body mass to sustain.
• Age: Affects metabolic rate as you grow older.

All of these factors are used in equations to calculate basal energy requirements. This ensures that the calorie intake recommended is tailored to suit the individual's needs and help maintain the basic functions of their body.

Metabolic Rate

The metabolic rate is the rate at which the body uses energy, measured in calories, to keep itself alive and functioning. Specifically, the Basal Metabolic Rate (BMR) represents the minimum number of calories needed when the body is at complete rest. It ensures that critical processes, such as heart beating and breathing, are supported.

Variables impacting the metabolic rate include:

• Genetics: Some people are naturally predisposed to burn more calories.
• Muscle Mass: More muscle leads to a higher BMR, as muscle tissue is active and requires more energy.
• Age: Typically, BMR decreases with age, leading to fewer calories burned at rest.

Understanding how BMR functions and how factors like weight, height, and age impact it is essential. This knowledge helps tailor lifestyle choices and dietary habits to maintain energy balance and overall health.

Age Impact

Age impacts how many calories are needed at rest. As age increases, metabolic processes tend to slow down. This is why in the given equations, the coefficient of age \( y \) is negative. It reflects the decrease in calorie needs with advancing age.

Important points to consider:

• Muscle Loss: With age, lean muscle mass naturally decreases, reducing calorie requirements.
• Decreased Activity Level: Often, older individuals have a more sedentary lifestyle, leading to lower energy needs.
• Metabolism: Aging can slow down metabolic activities, requiring fewer calories to function.

Thus, understanding how age affects basal metabolic rate helps shape dietary and exercise choices to maintain an optimal balance. It's why the equations use age as a minus factor, lowering the calorie count needed as one gets older.

Weight Impact

Weight has a significant impact on calorie requirements. In the equations for basal energy requirement, weight \( w \) is one of the main factors, with a positive coefficient. This means more weight generally results in a higher calorie need.

Key insights on weight impact include:

• Energy Demand: More body mass means more energy is needed to maintain essential bodily functions.
• Calorie Consumption: As weight increases, so does calorie consumption to support metabolic processes.
• Adjustment: Individuals with more weight may require tailored dietary plans to address their energy needs adequately.

So, weight directly influences how many calories the body burns, underscoring the need to consider weight in dietary planning for maintaining health and well-being.

Height Impact

Height is another critical factor influencing calorie needs. Taller individuals typically have more body mass and require more energy for basic physical functions, which explains the positive coefficient for height \( h \) in the basal energy requirement formulas.

Here's how height affects caloric needs:

• Body Surface Area: Taller people have a larger surface area, requiring more energy to regulate body temperature.
• Caloric Burn: Greater height often demands a higher calorie expenditure to maintain bodily functions.
• Growth and Development: Especially in younger individuals, more height can mean ongoing growth, necessitating higher energy needs.

Therefore, height plays a significant role in determining the basal metabolic rate, as computations adjust calorie recommendations based on an individual's stature.