Problem 64 - Calculus

Question

Give an example of: A formula representing the statement " \(q\) is inversely proportional to the cube root of \(p\) and has a positive constant of proportionality."

Step-by-Step Solution

Answer

An example formula is \( q = \frac{7}{\sqrt[3]{p}} \).

Step 1: Understanding the Statement

The problem states that \( q \) is inversely proportional to the cube root of \( p \). In mathematical terms, this means that \( q \) is equal to some constant divided by the cube root of \( p \).

Step 2: Expressing the Relationship

Since \( q \) is inversely proportional to the cube root of \( p \), we express this as \( q = \frac{k}{\sqrt[3]{p}} \), where \( k \) is the positive constant of proportionality.

Step 3: Considering the Proportionality Constant

Given that the constant of proportionality is positive, ensure \( k > 0 \). This satisfies the problem's condition that the constant is positive.

Step 4: Example Formula

An example of such a formula is \( q = \frac{7}{\sqrt[3]{p}} \), where \( 7 \) is a positive constant.

Key Concepts

Understanding Cube Roots

The cube root of a number, indicated by \( \sqrt[3]{x} \), helps determine a value that, when multiplied by itself three times, results in the original number. For example, the cube root of 8 is 2 because multiplying 2 by itself three times (2 x 2 x 2) equals 8. This operation is particularly useful when dealing with equations where one variable is related by the cube of another.
\( \sqrt[3]{x} \) operates differently than square roots because it relates to three dimensions.
It's beneficial to imagine a cube whose volume equals \( x \). The cube root represents the length of one side of that cube. For inverse relations, we focus on how changes in one dimension affect the entire cube, impacting values inversely in the case of proportional relationships.

Deciphering the Constant of Proportionality

In equations that describe a proportional relationship, the constant of proportionality plays a crucial role. This constant, often represented by \( k \), is a fixed number that influences the relationship between two variables.

• If you have two variables such as \( q \) and \( p \), the constant helps maintain the relationship defined in mathematical terms.
• For instance, if \( q = \frac{k}{\sqrt[3]{p}} \), \( k \) determines how changes in the cube root of \( p \) affect \( q \).

In this context, since \( k \) is specified as positive, it ensures that \( q \) remains positive for positive values of \( p \).
This constancy enables consistent application of the formula across various scenarios, making predictions and calculations reliable.

Formulating Mathematical Expressions

Mathematical expressions succinctly convey relationships between variables using numbers, symbols, and operations. They form the backbone of mathematics, allowing us to describe complex ideas in simpler terms.
For example, the expression \( q = \frac{7}{\sqrt[3]{p}} \) captures the inverse relationship we're discussing, where 7 is the constant of proportionality.

• Expressions combine constants, variables, and operators (like division and roots) to model real-world situations.
• By structuring an expression in this way, it clearly shows how \( q \) changes as \( p \) changes, specifically how \( q \) is inversely related to the cube root of \( p \).

This specific expression provides a toolkit for solving problems where understanding how one quantity diminishes as another grows is essential. Such expressions are essential for modeling behaviors across physics, economics, and beyond.