Problem 111 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I evaluated the formula \(d=\sqrt{1.5 h}\) for a value of \(h\) that resulted in a rational number for \(d\).

Step-by-Step Solution

Answer

The statement does not make sense.

Step 1: Understanding the Concept of Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both numerator and denominator are integers and the denominator is not zero. They can be positive, negative or zero.

Step 2: Understanding the Math Formula

In the formula \(d=\sqrt{1.5 h}\), \(d\) is determined by the square root of 1.5 times \(h\). Here, 1.5 is a rational number but it's not a perfect square. The square root of a product can be found by taking the square root of each factor separately, therefore, \(d\) can also be expressed as \(\sqrt{1.5} * \sqrt{h}\). We know that \(\sqrt{1.5}\) is an irrational number. Hence, if \(h\) is a rational number and especially if \(h\) is not a perfect square, the product is most likely to be an irrational number.

Step 3: Evaluating the Statement

Based on the above understanding, 'evaluating the formula \(d=\sqrt{1.5 h}\) for a value of \(h\) that resulted in a rational number for \(d\)', does not make sense since the product of an irrational number (i.e. \(\sqrt{1.5}\)) and potentially another irrational number (i.e. \(\sqrt{h}\) if \(h\) is not a perfect square) results in an irrational number, hence \(d\) cannot be a rational number.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of letters representing variables and numbers using arithmetic operations like addition, subtraction, multiplication, division, and exponentiation. For instance, in the expression \(d=\sqrt{1.5 h}\), \(d\) stands for distance, which is considered the dependent variable as its value is determined by \(h\), the independent variable representing height.

An essential skill in algebra is substituting variables with numbers and vice versa. When an expression involves a square root, like \(\sqrt{1.5 h}\), it incorporates arithmetic under the radical sign, where multiplication and rational numbers might come into play. Simplifying square roots can lead to either rational or irrational numbers, which significantly affects the outcome when evaluating the expression.

Demystifying Square Roots

Square roots are mathematical operations that answer the question: What number, when multiplied by itself, will produce this value? The square root of a number \(x\) is represented as \(\sqrt{x}\). A number like 1.5 is not a perfect square; hence its square root is not a whole number.

When dealing with square roots in equations, such as \(d=\sqrt{1.5 h}\), it's crucial to understand the nature of the numbers involved. If \(h\) were a perfect square (like 4, 9, 16, etc.), its square root would be a rational number. However, since 1.5 is not a perfect square, \(\sqrt{1.5}\) is an irrational number and multiplying it by any non-zero number will result in an irrational number, too, unless specific conditions are met that create a rational product.

Identifying Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction \(\frac{a}{b}\) where both \(a\) and \(b\) are integers and \(b\) is not zero. They have non-repeating, non-terminating decimal expansions. Examples include \(\pi\), \(e\), and the square root of any non-perfect square number like \(\sqrt{1.5}\).

Irrational numbers play a critical role in understanding the possible values of algebraic expressions. Since \(\sqrt{1.5}\) is irrational, multiplying it by another number, rational or not, will often result in another irrational number. Therefore, in the exercise, the assumption that evaluating \(d=\sqrt{1.5 h}\) for some value of \(h\) that results in a rational \(d\) does not align with the property that the product of a rational and an irrational number is generally irrational. This understanding is crucial for correctly interpreting the results of algebraic expressions involving square roots.