Problem 80 - Calculus: Early Transcendentals

Question

A model for the basal metabolism rate, in kcal/h, of a young man is \( R(t) = 85 - 0.18 \cos(\pi t/12) \), where \( t \) is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, \( \displaystyle \int^{24}_0 R(t) \, dt \), over a 24-hour time period?

Step-by-Step Solution

Answer

Total basal metabolism over 24 hours is 2040 kcal.

Step 1: Understanding the Function

The function given is a model for the basal metabolism rate. The function is \( R(t) = 85 - 0.18 \cos(\pi t/12) \), where \( t \) is the time in hours from 5:00 AM. This represents how metabolism changes throughout a day.

Step 2: Set Up the Integral

We need to integrate \( R(t) \) over one full day, which means calculating \( \int_{0}^{24} (85 - 0.18 \cos(\pi t/12)) \, dt \). This will provide the total metabolism over 24 hours.

Step 3: Integrate Constant and Cosine Terms Separately

The integral \( \int_{0}^{24} 85 \, dt \) is straightforward as \( 85 \) is a constant. For the cosine term, use the integration rule: \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \).

Step 4: Calculate the Constant Term

Integrate the constant term:\[ \int_{0}^{24} 85 \, dt = 85 \times (24 - 0) = 2040. \]

Step 5: Calculate the Cosine Term

Integrate the cosine term:\[ \int_{0}^{24} -0.18 \cos(\pi t/12) \, dt = -0.18 \left[ \frac{12}{\pi} \sin(\pi t/12) \right]_{0}^{24}. \]

Step 6: Evaluate the Cosine Integral

Substitute the limits into the integrated form:\[ -0.18 \left[ \frac{12}{\pi} (\sin(2\pi) - \sin(0)) \right] = -0.18 \times \frac{12}{\pi} \times 0 = 0. \] Since the sine function evaluates to 0 at both ends, the entire term simplifies to 0.

Step 7: Sum Both Parts

Combine the results from the previous steps:\[ 2040 + 0 = 2040. \]Thus, the total basal metabolism over a 24-hour period is 2040 kcal.

Key Concepts

Basal Metabolism

Basal metabolism refers to the amount of energy expended while at rest in a neutrally temperate environment. It represents the minimum amount of energy required to keep the body functioning, including vital organs like the heart, lungs, and other essential systems.
Thus, it's an essential consideration for understanding energy needs over a typical day. Understanding it through the lens of calculus allows us to model how it changes over time and calculate the total energy expenditure.

Trigonometric Integration

Trigonometric integration involves finding the integrals of trigonometric functions. In this exercise, the trigonometric function of interest is the cosine function.
Integrating cosine functions is a vital skill in calculus since they frequently appear in periodic functions and real-world models like the basal metabolism rate.

• The integral of the cosine function \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \).
• This formula is applied to the exercise’s function, allowing us to address how metabolism varies with time through cosine adjustments.

Definite Integral

Definite integrals calculate the net area under a curve over a specific interval from \(a\) to \(b\).
In practical terms, a definite integral gives us the total amount of something – like metabolism rate – accumulated over this time period.

• For this particular problem, we're evaluating \( \int_{0}^{24} R(t) \, dt \) to determine how much energy is expended in a day.
• The process includes splitting the integral into manageable parts: a constant and a cosine function, then calculating each component separately.

Calculus Applications

Calculus, especially integral calculus, is widely used to solve real-world problems. One of its main applications is computing accumulated quantities, such as the total basal metabolism over time, as shown in this exercise.
Through integration, we can predict and analyze various physical phenomena and processes.

• By applying calculus to the model \( R(t) \), we found the daily energy expenditure accurately.
• This is just one example of how calculus helps in fields such as biology, physics, and even economics.