Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Applications Involving Powers The volume of a cube of side \(35.8 \mathrm{cm}\) (Fig. \(1-6\) ) is \((35.8)^{3} .\) Evaluate this volume.

Short answer

The volume of the cube with a side length of 35.8 cm, when rounded to three significant digits, is \( 45900 \, \mathrm{cm}^3 \) .

Step-by-step solution

Understanding the Cube Volume

The volume of a cube is calculated by raising the length of one side to the power of three. Given the side length of the cube is 35.8 cm, the volume formula for the cube is \( V = s^3 \) where \( V \) is the volume and \( s \) is the side length.

Calculate the Volume

Substitute the given side length into the volume formula: \( V = (35.8 \, \mathrm{cm})^3 \) and calculate the value.

Consider Significant Digits

The given side length has three significant digits. Therefore, the final answer should also be rounded to three significant digits to properly represent the precision of the measurement.

Evaluate the Expression

Evaluating the expression \( (35.8)^3 \) yields \( 35.8 \times 35.8 \times 35.8 \) which equals approximately \(45912.792 \, \mathrm{cm}^3 \) before rounding.

Round to Proper Significant Digits

Round the calculated volume, 45912.792, to three significant digits. The rounded volume of the cube is \( 45900 \, \mathrm{cm}^3 \) (rounded to the nearest hundred because of the decimal place in the original number).

Key concepts

Volume of a Cube

When we talk about the volume of a cube, we're referring to the amount of space enclosed within it. The cube is one of the simplest 3-dimensional shapes, and because of its symmetrical properties, its volume is easy to calculate. The volume formula for a cube is defined as the length of a side raised to the power of three, mathematically expressed as \( V = s^3 \) where \( V \) represents volume and \( s \) represents the side length of the cube.

For instance, if a cube has a side length of \(35.8 \text{cm}\), its volume would be calculated as \((35.8 \text{cm})^3\). Understanding this concept is crucial because it not only applies to theoretical mathematics but also to real-life scenarios where one might need to determine the space an object would occupy, for example, in packaging or container design.

Exponentiation in Mathematics

Exponentiation is a form of mathematical shorthand for expressing repeated multiplication of the same factor. The number being multiplied is known as the base, and the number of times the base is multiplied by itself is the exponent. In our cube volume example, the side length (base) is raised to the power of three (exponent), indicated by \((35.8)^3\).

This tells us to multiply the side length by itself two additional times. Exponentiation simplifies expressions and makes it easier to perform calculations involving powers. Being comfortable with exponentiation is essential for students as it is widely used in various math concepts such as geometric calculations, scientific notation, and growing patterns.

Rounding to Significant Figures

Significant figures are a way of expressing precision in measured quantities. When we round numbers, we want to keep the most meaningful digits, which are known as significant figures. In a measurement, all non-zero digits are significant, as well as any zeros between them. There might be ambiguity with zeros at the beginning or end of a number, so rules for those depend on the number's decimal placement or if it's in scientific notation.

In the volume of a cube, our calculated result of \(45912.792 \text{cm}^3\) should be rounded to three significant figures because our input (the side length) was given to three significant digits (35.8). This yields a final answer of \(45900 \text{cm}^3\). This process ensures the answer reflects the precision of the input data and does not imply a false level of accuracy.

Mathematical Problem Solving

Mathematical problem solving is not just about getting the right answer; it's about understanding the process and applying it to different scenarios. A good problem-solving approach involves a clear understanding of the problem, devising a plan, executing that plan, and then reflecting on the result to understand if it makes sense.

In our exercise, the initial step was to comprehend the formula for the volume of a cube. Following that, we substituted the given side length value into the formula and performed the calculations while considering the rules of exponentiation. The final and crucial phase was to apply the concept of significant figures to round our answer appropriately, thus completing the problem-solving process whilst maintaining the mathematical integrity of our solution.