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Chapter 15: Problem 33

Find the decimal approximation rounded to the nearest thousandth. $$\sqrt{240}$$

Short Answer

Expert verified
The square root of 240, rounded to the nearest thousandth, is \(15.492\).

Step by step solution

01

Calculating Square Root

First, calculate the square root of 240. This can be done with a calculator. The square root of 240 is approximately \(15.4919333848\)
02

Rounding to Nearest Thousandth

To round to the nearest thousandth, look at the fourth digit after the decimal point. If that digit is 5 or more, the third digit after the decimal point increases by 1. In this case, since the fourth digit after the decimal point is 9 which is more than 5, the third digit after the decimal point, 9, becomes 9+1=10, we carry one to preceding 1 and it becomes 2, replacing 1, hence, we get \(15.492\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 gives 16. Square roots are essential in various fields of mathematics, especially in geometry and algebra. They help in understanding distances, solving quadratic equations, and in many other applications.

When you encounter a problem like finding the square root of 240, you are essentially looking for a number that, when squared, yields 240. Since 240 is not a perfect square (like 16 or 25), its square root is not an integer. This requires us to use a calculator to get a decimal approximation.
Rounding
Rounding is the process of reducing the number of digits in a number while maintaining a value that is close to the original number. It helps make numbers easier to work with in everyday calculations and estimations.

In mathematics, different rules exist for rounding. A standard rule is to look at the next digit in the sequence to decide whether to round up or down. For example, when rounding 15.4919333848, you observe the fourth digit after the decimal point. If this digit is 5 or higher, you increase the last digit you want to keep by one. Otherwise, you leave it the same. Rounding helps simplify results, especially when dealing with long decimals, as in our exercise.
Thousandth Place
The thousandth place is a position in a decimal number, three places to the right of the decimal point. This level of precision is often necessary in scientific calculations, measurements, and other areas where exactitude is important.

When rounding a number like 15.4919333848 to the nearest thousandth, you maintain up to three decimal places. This means you will retain 15.492 as your answer, reflecting the precision needed, but not more digits than are practical for your purpose. Maintaining precision often depends on what you're using your result for, and the thousandth place offers a good balance between accuracy and simplicity.
Calculator Use
Using a calculator to find a decimal approximation is a straightforward and effective method, especially for more complex numbers like non-perfect squares. When computing the square root of 240, simply entering \( \sqrt{240} \) into a calculator provides you with the full decimal approximation quickly.

Calculators can also assist with the rounding process. Some calculators have settings that automatically round to a specified decimal place, saving time and reducing manual errors. By relying on this technology, students can focus more on understanding the core mathematical concepts rather than getting bogged down in calculation details.

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Most popular questions from this chapter

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