Problem 56 - Calculus for Biology and Medicine

Question

Sketch each scaling relation (Niklas, 1994). Suppose that a cube of length \(L\) and volume \(V\) has mass \(M\) and that \(M=0.35 V\). How does the length of the cube depend on its mass?

Step-by-Step Solution

Answer

The cube's length \(L\) depends on its mass \(M\) as \(L = \left(\frac{M}{0.35}\right)^{1/3}\).

Step 1: Define the Volume Formula

A cube with length \(L\) has a volume \(V\), which can be expressed in terms of its side length \(L\) as \(V = L^3\).

Step 2: Substitute for Volume in Mass Equation

We know that the mass \(M\) is given by the equation \(M = 0.35V\). Substitute \(V = L^3\) into this equation, obtaining \(M = 0.35L^3\).

Step 3: Solve for Length \(L\) in Terms of Mass \(M\)

To express \(L\) in terms of \(M\), solve the equation \(M = 0.35L^3\) for \(L\). First, divide both sides by 0.35 to get \(L^3 = \frac{M}{0.35}\). Then take the cube root to solve for \(L\), yielding \(L = \left(\frac{M}{0.35}\right)^{1/3}\).

Key Concepts

Volume of a Cube

When you think of a cube, it's important to visualize it as a three-dimensional shape with equal sides. Each side of this cube is denoted by the length \(L\). So, when calculating its volume, which is essentially the space inside the cube, we use the formula:

• \(V = L^3\)

This formula comes from multiplying the length by itself twice (since all sides are equal), providing us with the total cubic units of space inside. The simplicity of the cube's geometry makes this calculation quite straightforward. Understanding how to determine the volume is crucial because it ties into many other properties, like the cube's mass and how it scales with other measurements. Every time you measure the volume using the cube's side length, remember that it's a repeated multiplication tailored for cubes.

Mass-Volume Relationship

The connection between a cube's volume and its mass is fascinating and important for scaling relations. In the given problem, this relationship is expressed as:

• \(M = 0.35V\)

The constant 0.35 here signifies that the mass is a portion of the volume. Proportion values such as 0.35 imply that we can understand how dense or light a cube is compared to its volume. In practical terms, if you know the volume, using the constant 0.35, you'll readily find the mass. This constant helps simplify real-world situations where materials might change, offering a reliable way to compare similar cubic shapes. It's essential to know that this expression assures that as the volume changes, the mass will change in a proportional manner, maintaining that specific ratio at all times.

Cube Root Calculation

Finding the cube root is an essential mathematical skill, especially when dealing with volumes and side lengths of cubes. Suppose you need to determine the length \(L\) from a given mass \(M\). To do so, you'll use the equation derived from the mass-volume relationship:

• First, solve for \(L^3\) in \(M = 0.35L^3\) by rearranging it to \(L^3 = \frac{M}{0.35}\).
• Then, find the cube root: \(L = \left(\frac{M}{0.35}\right)^{1/3}\).

The cube root provides the side length \(L\) because it reverses the cubing process that initially gave us the volume. The steps to find the cube root involve dividing the mass by the constant 0.35 and then taking the cube root of the resulting number. Cube roots can seem challenging at first, but they become manageable with practice and understanding. By computing this correctly, you can convert between the mass relationship to find practical geometry solutions for cubes in a variety of applications.