Problem 105 - Algebra and Trigonometry with Analytic Geometry

Question

Calorie requirements 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: $$ \begin{aligned} &C_{f}=66.5+13.8 w+5 h-6.8 y \\ &C_{m}=655+9.6 w+1.9 h-4.7 y \end{aligned} $$ (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. (b) Discuss why, in both formulas, the coefficient for \(y\) is negative but the other coefficients are positive.

Step-by-Step Solution

Answer

Female: 1413.6 calories; Male: 1617.5 calories. Age reduces calorie needs, while weight and height increase them.

Step 1: Determine Basal Energy for Female

To find the basal energy requirement for the female, we use the formula: \[ C_{f} = 655 + 9.6w + 1.9h - 4.7y \] Substitute the given values: weight \( w = 59 \) kg, height \( h = 163 \) cm, and age \( y = 25 \) years.\[ C_{f} = 655 + 9.6(59) + 1.9(163) - 4.7(25) \] Calculate: \( 9.6 \times 59 = 566.4 \), \( 1.9 \times 163 = 309.7 \), and \( 4.7 \times 25 = 117.5 \).\[ C_{f} = 655 + 566.4 + 309.7 - 117.5 \] = 1413.6 So, the basal energy requirement for the female is 1413.6 calories.

Step 2: Determine Basal Energy for Male

Use the formula for males:\[ C_{m} = 66.5 + 13.8w + 5h - 6.8y \]Substitute the given values: weight \( w = 75 \) kg, height \( h = 178 \) cm, and age \( y = 55 \) years.\[ C_{m} = 66.5 + 13.8(75) + 5(178) - 6.8(55) \]Calculate: \( 13.8 \times 75 = 1035 \), \( 5 \times 178 = 890 \), and \( 6.8 \times 55 = 374 \).\[ C_{m} = 66.5 + 1035 + 890 - 374 \] = 1617.5 So, the basal energy requirement for the male is 1617.5 calories.

Step 3: Discussion of Coefficients

The coefficient for the age \( y \) is negative in both formulas because as people age, their basal metabolic rate tends to decrease, which leads to a lower calorie requirement. The positive coefficients for weight and height indicate that a larger mass and stature require more energy to maintain basic bodily functions due to the greater amount of body tissue that needs to be sustained.

Key Concepts

Calorie Calculation

Understanding how to calculate the number of calories you need daily is essential. Calories are the units of energy that fuel our bodies. For those looking to maintain their body functions without gaining or losing weight, calculating your calorie needs is the first step. This is often referred to as the basal energy requirement.
Consequently, calculating your calories involves more than just a singular number. It requires considering various factors such as your sex, weight, height, and age. The formulas for calculating calories differ for males and females due to physiological differences. For example:

• The formula for females: \( C_{f}=655+9.6w+1.9h-4.7y \)
• The formula for males: \( C_{m}=66.5+13.8w+5h-6.8y \)

With these factors plugged in, you can find your basal energy requirement that tells you the minimum calories needed to keep your body functioning when at rest.

Metabolic Rate

Metabolic rate is a term that often comes up when discussing calorie needs. It refers to the rate at which your body uses energy or burns calories. A key aspect of the metabolic rate is the Basal Metabolic Rate (BMR), which measures the energy you need at rest to maintain vital bodily functions, like breathing and circulation.
Several factors influence your metabolic rate:

• Weight: More body mass often demands more calories because of the greater amount of tissue that needs sustaining.
• Height: Taller individuals typically have a higher BMR because they have more body surface area.
• Age: As you age, your metabolic rate generally decreases, which is why the coefficient for age is negative in calorie formulas.

Understanding your metabolic rate gives you an insight into how your body processes energy. It's a fundamental concept in calculating how many calories your unique body requires each day.

Energy Expenditure

Energy expenditure is all about how the body expends energy or burns calories. This encompasses basal energy requirements, physical activity, and even digestion. Understanding how energy expenditure works helps you balance what you eat with what you burn to maintain or alter your weight.
The components of energy expenditure include:

• Basal Metabolic Rate (BMR): The energy used at rest, sustaining body functions.
• Physical Activity: The calories burned during exercise and everyday movements.
• Thermic Effect of Food (TEF): The energy used to digest and metabolize food.

By understanding these components, you can tailor your lifestyle to meet your energy needs, whether for maintaining health, losing weight, or building muscle.

Nutrition Science

Nutrition science is the study of how the body uses food and nutrients to maintain health and function. It's an evolving field focusing on how diet impacts our wellbeing. With a solid grasp of nutrition science, you can make informed decisions about your diet and understand its impacts on your health.
At the heart of nutrition science are macronutrients: carbohydrates, proteins, and fats. These provide the calories that give us energy. Here are some quick basics:

• Carbohydrates: The primary energy source, particularly for quick energy and brain function.
• Proteins: Essential for growth, repair, and maintenance of tissues.
• Fats: Vital for energy storage, insulation, and protecting organs.

By studying the science of nutrition, you can better understand how different foods affect your energy levels, mood, and overall health. This knowledge allows you to craft a balanced diet that meets your caloric and nutrient needs effectively.