Question

Middle-aged men require a base level of 900 Calories per day plus an additional 12 Calories per kilogram of body mass per day. Young adult women require a base level of 500 Calories per day, plus 15 Calories per kilogram of body mass per day. At what mass do middle-aged men and young adult women have the same caloric needs? (A) 26 kg (B) 67 kg (C) 133 kg (D) 266 kg

Short answer

The mass at which both have the same caloric needs is approximately 133 kg (Option C).

Step-by-step solution

- Define caloric needs formulas

First, write the formulas for the caloric needs of both middle-aged men and young adult women.\[ C_m = 900 + 12m \]\[ C_w = 500 + 15w \]

- Set the caloric needs equal

To find the mass where both middle-aged men and young adult women have the same caloric needs, set the two equations equal to each other.\[ 900 + 12m = 500 + 15w \]

- Simplify the equation

Since we are interested in the mass where both middle-aged men and young adult women have the same caloric needs, denote this mass as m, and recognize that w = m in this context. Thus, rewrite the equation as: \[ 900 + 12m = 500 + 15m \]

- Solve for m

Rearrange the equation to isolate m. Subtract 12m from both sides:\[ 900 = 500 + 3m \]Then, subtract 500 from both sides: \[ 400 = 3m \]Finally, divide both sides by 3 to find m: \[ m = \frac{400}{3} \approx 133.33 \]

- Determine the closest provided option

The calculation gives a value of approximately 133.33 kg. The closest provided option is 133 kg.

Key concepts

Caloric Intake

Caloric intake refers to the total number of calories consumed through eating and drinking. It is essential for providing energy to our bodies for various activities and bodily functions. Calculating caloric intake helps people manage their diet, nutrition, and overall health. In our exercise, we are concerned with how middle-aged men and young adult women calculate their daily caloric needs based on different base levels and factors.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It represents numbers and relationships in equations. In this exercise, algebra is used to express the caloric needs of middle-aged men and young adult women in formula form. Understanding how to translate word problems into algebraic equations is essential for solving mathematical problems efficiently.

Equation Solving

Equation solving involves finding the value of unknown variables that satisfy the given mathematical equation. In this case, we need to determine the mass at which the caloric needs of both middle-aged men and young adult women are equal. By setting their caloric needs equations equal, we derive a singular equation. Simplifying and manipulating this equation algebraically then allows us to solve for the unknown variable representing mass.

Daily Caloric Requirements

Daily caloric requirements are the total number of calories a person needs in a day to maintain their current body weight and health. It varies by age, gender, weight, and activity level. For middle-aged men, the base is 900 calories plus an additional 12 calories per kg of body mass. For young adult women, it is 500 calories plus an additional 15 calories per kg. Understanding daily caloric requirements is vital for maintaining a balanced diet and health.