Question

Vocabulary Terms - Define/Illustrate
Interpreting Irrational answers for side lengths of squares and cubes.
Interpreting Irrational answers for time and money

Answer 1

Certainly! Let's define and illustrate the two vocabulary terms you mentioned: **Interpreting Irrational Answers for Side Lengths of Squares and Cubes** and **Interpreting Irrational Answers for Time and Money**.

### 1. Interpreting Irrational Answers for Side Lengths of Squares and Cubes

**Definition:**
When calculating the side lengths of squares or cubes, we may come across irrational numbers. An irrational number cannot be expressed as a simple fraction (i.e., it cannot be written as a/b, where a and b are integers). Common examples include \(\sqrt{2}\), \(\sqrt{3}\), and \(\pi\). In geometric contexts, irrational answers can arise in cases where the side length is derived from the area or volume, leading to non-integer values.

**Illustration:**
- **Example 1: Finding the Side Length of a Square**
- If the area of a square is \(2\), to find the side length \(s\), we use the formula:
\[
s^2 = 2
\]
Taking the square root gives us:
\[
s = \sqrt{2} \approx 1.41
\]
- This value is irrational, but it represents the exact side length necessary to achieve a square area of 2 square units.

- **Example 2: Finding the Side Length of a Cube**
- If the volume of a cube is \(27\), then the side length \(s\) can be found using:
\[
s^3 = 27
\]
Taking the cube root gives:
\[
s = 3
\]
- Here, 3 is rational. However, if the volume had been \(32\), then:
\[
s^3 = 32 \quad \Rightarrow \quad s \approx \sqrt[3]{32} \approx 3.17 \text{ (irrational)}
\]

### 2. Interpreting Irrational Answers for Time and Money

**Definition:**
In some cases, calculations concerning time or money can yield irrational numbers. Although time and money are typically expressed in finite units (like hours, minutes, dollars, and cents), certain mathematical operations (such as interest calculations or proportions) may produce irrational results. Understanding these values is important when making practical applications or estimations.

**Illustration:**
- **Example 1: Time Calculation**
- Suppose an intrinsic rate of return is calculated and it results in an irrational number, such as \(5.236\) hours for some investment or task. While the raw number is irrational, it suggests that it may take approximately \(5\) hours and \(14\) minutes to complete the task. The actual task would not take precisely that time, but closed approximations become necessary for practical scheduling.

- **Example 2: Money Calculation**
- If a bank offers an annual interest rate which mathematically calculates a compounded return, you might end up with an amount like \(\$105.50 + \sqrt{2} \approx \$106.41\). Here, the irrational component does not directly relate to a currency but gives insight into return fluctuations. When interpreting this, one would round it to the nearest cent (\$106.41), as actual currency transactions require rational numbers for practicality.

In both cases, while dealing with irrational results, the context dictates how these values are applied or approximated for clear communication and practical use.