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

Human's circulation time is approximately 46.25 seconds.

Step 1: Understanding the Proportionality

We are given that the circulation time, \(T\), is proportional to the fourth root of the body mass, \(B\). In mathematical terms, this can be expressed as \(T = k \cdot B^{1/4}\), where \(k\) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

We know that when \(B = 5230\) kg, the circulation time \(T = 148\) seconds. Plug these values into the formula to find \(k\):\[148 = k \cdot (5230)^{1/4}\]We solve for \(k\) by dividing both sides by \((5230)^{1/4}\):\[k = \frac{148}{(5230)^{1/4}}\]Use a calculator to calculate \((5230)^{1/4}\), approximately 8.45, so:\[k \approx \frac{148}{8.45} \approx 17.52\]

Step 3: Calculating Circulation Time for Human

Now that we know \(k \approx 17.52\), we can find the circulation time for a human with a body mass of 70 kg using the formula: \[T = 17.52 \cdot (70)^{1/4}\]Calculate \((70)^{1/4}\), approximately 2.64, so:\[T \approx 17.52 \cdot 2.64 \approx 46.25\] seconds.

Key Concepts

Proportional Relationships

Proportional relationships are a fundamental concept in mathematics where two quantities are related in such a way that their ratio remains constant. This means if you multiply one quantity by a certain number, the other quantity is multiplied by the same number to maintain the ratio.
In the context of our exercise, the circulation time of a mammal is proportional to the fourth root of its body mass. This can be expressed using the formula \( T = k \cdot B^{1/4} \), where \( T \) is the circulation time, \( B \) is the body mass, and \( k \) is the constant of proportionality.
Understanding proportional relationships helps in predicting how changes in one quantity affect another.

• If the body mass increases, the circulation time will also increase, but at a specific rate determined by the constant \( k \).
• Conversely, a decrease in body mass will lead to a decrease in circulation time.

Recognizing proportionality in problems helps simplify calculations and predict outcomes.

Body Mass Calculations

Calculating body mass, especially concerning its relationship with biological processes, is crucial. In our exercise, body mass is important because it directly affects circulation time through a specific mathematical relationship.
Body mass in equations can often be raised to a power, like in our formula \( B^{1/4} \). Here, the body mass is taken to the one-quarter power or the fourth root, which serves as an adjustment to better match real-world biological phenomena.
Understanding how to work with body mass in formulas is important:

• For example, plugging in the body mass of an elephant at 5230 kg helps find \( k \), the constant of proportionality, by solving \( T = k \cdot (5230)^{1/4} \).
• This process involves using a calculator to find the root and solve for the unknown.

Proper calculations ensure accurate predictions of properties like circulation time, which depend heavily on an accurate understanding of body mass.

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), is a crucial element in equations involving proportional relationships. It acts as a fixed multiplier that adjusts the relationship between two proportional quantities.
In our problem, \( k \) is determined by using known values: the body mass and circulation time of an elephant. Given \( 5230 \) kg and a circulation time of \( 148 \) seconds, the formula \( T = k \cdot B^{1/4} \) allows us to solve for \( k \) as follows:

• Calculate the fourth root of the body mass: \( (5230)^{1/4} \).
• Substitute \( T = 148 \) and \( (5230)^{1/4} \) back into the formula and solve for \( k \).
• This results in \( k = 17.52 \) after performing the necessary division.

This constant is then used for future calculations, such as determining circulation times for other body masses, ensuring consistency and accuracy in predictions.