Problem 25 - Applied Calculus

Question

The circulation time of a mammal (that is, the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal. (a) Write a formula for the circulation time, \(T\), in terms of the body mass, \(B\). (b) If an elephant of body mass 5230 kilograms has a circulation time of 148 seconds, find the constant of proportionality. (c) What is the circulation time of a human with body mass 70 kilograms?

Step-by-Step Solution

Answer

The circulation time formula is \( T = k \cdot B^{1/4} \). The constant of proportionality \( k \) is approximately 12.67. A human with 70 kg body mass has a circulation time of about 33.64 seconds.

Step 1: Understanding Proportional Relationships

In the problem statement, it is mentioned that the circulation time, \( T \), is proportional to the fourth root of the body mass, \( B \). This can be expressed mathematically as \( T = k \cdot B^{1/4} \), where \( k \) is the constant of proportionality.

Step 2: Establish the Formula

We can now establish the formula for \( T \) in terms of \( B \). Hence, the formula is: \[ T = k \cdot B^{1/4} \]

Step 3: Use Given Data to Find the Constant of Proportionality

Given that for an elephant with a body mass of 5230 kg, the circulation time \( T \) is 148 seconds. We substitute these values into the formula: \[ 148 = k \cdot 5230^{1/4} \]. Calculate \( 5230^{1/4} \), then solve for \( k \): \( k = \frac{148}{5230^{1/4}} \).

Step 4: Calculate the Fourth Root and Solve for k

First, calculate \( 5230^{1/4} \) using a calculator, which approximately equals 11.68. Then, \( k = \frac{148}{11.68} \). Calculating this gives \( k \approx 12.67 \).

Step 5: Apply the Formula for a Human

Now, with \( k \approx 12.67 \), use this to find the circulation time for a human with a body mass of 70 kg: \[ T = 12.67 \times 70^{1/4} \]. Calculate \( 70^{1/4} \) first.

Step 6: Calculate for a Human

Find \( 70^{1/4} \), which is approximately 2.655. Then, substitute back to find circulation time: \( T = 12.67 \times 2.655 \). This equals approximately 33.64 seconds.

Key Concepts

Fourth Root

The concept of the fourth root is fundamental in this problem. In mathematics, the fourth root of a number is a value that, when raised to the power of four, equals the original number. In simpler terms, if you multiply the fourth root by itself four times, you will get back to the number you started with.
To express this mathematically, the fourth root of a number is denoted as \( B^{1/4} \). This means if you have a body mass \( B \), and you're looking for its fourth root, you are essentially solving for \( B^{1/4} \).
This concept is applied in the context of the circulation time for mammals. Here, the circulation time is proportional to the fourth root of the body mass, showing a relationship between how quickly blood circulates and the size of the body in exponential terms.

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), is a factor that describes the relationship between two proportional quantities. It remains constant even as the individual involved variables change.
In this exercise, we use the formula \( T = k \cdot B^{1/4} \) to illustrate how circulation time \( T \) relates to body mass \( B \). The constant \( k \) helps convert the body mass’s fourth root into the actual circulation time in seconds.
To find the constant \( k \), we use known values, which, in this case, are the elephant's body mass and circulation time. By inserting these into the formula and calculating, \( k \) can be determined, providing a crucial connection that helps solve the problem for different body masses.

Body Mass

Body mass is the total mass of an organism and is a critical factor in this exercise. It is given in kilograms and directly influences the circulation time due to its proportional relationship with that time.
It's important to recognize how variations in body mass affect this time. In mammals, as body mass increases, the circulation time increases as well since it is tied to the fourth root of the mass.
Body mass plays a role in understanding the biological design and efficiency of organisms. By calculating things like circulation time using body mass, researchers can make insights about how different body sizes impact physiological functions. For example, larger animals typically have slower circulation times due to the bigger volume of blood that needs to be circulated, as illustrated by the examples of an elephant and human in this problem.