Problem 39 - Algebra Form and Function

Question

The blood mass of a mammal is proportional to its body mass. A rhinoceros with body mass 3000 kilograms has blood mass of 150 kilograms. Find a formula for the blood mass of a mammal in terms of the body mass and estimate the blood mass of a human with body mass 70 kilograms.

Step-by-Step Solution

Answer

Answer: The estimated blood mass of a human with a body mass of 70kg is 3.5 kg.

Step 1: Calculate proportionality constant for the rhinoceros

Recall that for a proportional relationship, we have: BloodMass = k * BodyMass where "k" is the proportionality constant. We are given the blood mass (150kg) and body mass (3000kg) of a rhinoceros. Plugging these values into the equation, we can solve for k: 150 = k * 3000 Now, we can isolate k by dividing both sides by 3000: k = 150 / 3000 Calculating the value of the constant 'k': k = 0.05

Step 2: Derive the general formula for blood mass of a mammal

Now that we have found the proportionality constant (k = 0.05), we can derive the general formula for the blood mass of a mammal in terms of its body mass. The formula is: BloodMass = k * BodyMass Since we know the proportionality constant, we can write the formula as: BloodMass = 0.05 * BodyMass

Step 3: Estimate the blood mass of a human with a body mass of 70kg

We are given the body mass of a human, which is 70kg. Using the general formula for blood mass that we derived in step 2, we can estimate the blood mass of the human: BloodMass = 0.05 * BodyMass BloodMass = 0.05 * 70 Calculating the blood mass: BloodMass = 3.5 kg Therefore, the estimated blood mass of a human with a body mass of 70kg is 3.5kg.

Key Concepts

Blood Mass Calculation

In order to understand how the blood mass of mammals is calculated, it's important to recognize the idea of proportional relationships. This means the blood mass of a mammal varies directly with its body mass. In our exercise, we consider a rhinoceros as an example, where 150 kg of blood corresponds to a body mass of 3000 kg. To make calculations easier, we need to find a mathematical relationship between these two values.

The initial step is finding the **proportionality constant** (k) that links the blood mass and body mass. Once this constant is determined, applying it to other mammals becomes straightforward. In this scenario, we will use the equation:

• BloodMass = k * BodyMass

This formula allows you to predict blood mass for any given body mass by simply replacing the existing values.

Proportionality Constant

In scenarios where a linear or proportional relationship exists, identifying the proportionality constant 'k' is crucial. The constant helps to maintain consistency in predictions or calculations. Let's explore how this constant is derived and its function in our problem.

Calculating the constant

To calculate 'k' for the rhinoceros, we took these steps:

• We know the blood mass and body mass: 150 kg and 3000 kg, respectively.
• Insert these values into the equation: 150 = k * 3000.
• Solving for 'k' by dividing both sides by the body mass gives: k = 150 / 3000.

This results in a proportionality constant k = 0.05. This value of 'k' signifies that 5% of the total body mass comprises the blood mass.

Importance of the constant

The constant is essential because it translates to different mammals, offering a quick reference for estimating blood volumes based on body weights. It's important to realize that this constant is derived from specific cases and should be applied where similar conditions are assumed.

Mathematical Formula Derivation

Deriving the mathematical formula from the observed data is fundamental to predicting blood mass in different mammals. Understanding this derivation can help you apply the formula to varying contexts, improving both accuracy and efficiency in estimations.

From observation to equation

After deriving the constant 'k', we utilize it to form a general formula. Since we've established that the proportional relationship is true for our existing data, we write:

• BloodMass = k * BodyMass
• where k = 0.05.

This leads us to the refined formula:

• BloodMass = 0.05 * BodyMass

Applying this formula to new examples, like estimating the blood volume for a human weighing 70 kg, becomes straightforward:

• BloodMass = 0.05 * 70
• Calculating this gives us a blood mass of 3.5 kg.

Using the derived formula, students can assess blood mass accurately given any mammal's body weight, as long as they understand the underlying assumptions of proportionality.