Problem 52 - Precalculus with Limits

Question

In Exercises 49-58, find a mathematical model for the verbal statement. \(h\) varies inversely as the square root of \(s\).

Step-by-Step Solution

Answer

The mathematical model that corresponds to the statement 'h varies inversely as the square root of s' is \(h=k/(\sqrt{s})\).

Step 1: Identify the nature of variation

First, understand that the phrase 'varies inversely' means the relationship can be represented by the mathematical form \(y=k/x\), where \(y\) is the variable that varies, \(x\) is the variable it varies with, and \(k\) is the constant of variation.

Step 2: Apply the variables given in the exercise

In this case, \(h\) varies inversely as the square root of \(s\). Therefore, the form \(y=k/x\) becomes \(h=k/(\sqrt{s})\).

Step 3: Simplify if necessary

There is no need to simplify further, as \(h=k/(\sqrt{s})\) is already a simplified and correct mathematical model of the given verbal statement.

Key Concepts

Mathematical Modeling

Mathematical modeling is a process that involves creating mathematical representations of real-world situations. This enables us to predict and analyze complex phenomena using mathematical language and tools. In the context of the inverse variation problem, mathematical modeling translates a verbal statement into an algebraic equation that captures the essence of the relationship between the variables involved.

In our example, the verbal statement is that 'h varies inversely as the square root of s.' To model this situation, we use the concept of inverse variation. Inverse variation indicates that when one variable increases, the other decreases at a rate that maintains a consistent product, known as the constant of variation. The resulting model is not just an abstract representation; it's a tool that allows us to compute one variable when the other is known, making predictions and problem-solving possible in a range of applications from physics to economics.

Constant of Variation

The constant of variation is a key element in understanding inverse variation relationships. It's the steady number that ties two inversely varying quantities together. Regardless of the values that the variables take on, the product of one variable and a specific function of the other remains equal to the constant of variation.

In the equation \(h = \frac{k}{\sqrt{s}}\), 'k' represents the constant of variation. It's vital to note that 'k' remains fixed while the variables 'h' and 's' change. This constancy is the hallmark of inverse variation; even as one variable increases, the other must adjust accordingly to keep the product with the constant the same. It’s this concept that allows us to create reliable models that mirror real-world behavior.

Square Root Function

The square root function, symbolized by \(\sqrt{x}\), is fundamental in both basic and advanced mathematics. It represents one of the two equal factors of a non-negative number. When we say that 'h varies inversely as the square root of s' in our problem, the square root function plays a crucial role in determining the nature of the variation.

Mathematically, the square root function introduces a non-linear element to the inverse relationship. This means that as 's' grows, the decrease in 'h' is not by a straight proportional rate but rather in accordance with the outcome of the square root of 's'. Understanding how the square root affects the rate of variation is essential when constructing and interpreting mathematical models involving this function.