Problem 41 - Prealgebra

Question

If \(x=\sqrt{5000},\) what is the value of \(x\) to the nearest tenth?

Step-by-Step Solution

Answer

The value of \(x\) to the nearest tenth is 70.7.

Step 1: Estimate the Square Root

First, identify numbers whose squares are close to 5000. Notice that \(70^2 = 4900\) and \(80^2 = 6400\), so \(\sqrt{5000}\) is between 70 and 80.

Step 2: Narrow Down Further

Since \(70^2 = 4900\) is closer to 5000 than \(80^2 = 6400\), \(\sqrt{5000}\) is slightly above 70. Try \(72\): \[ 72^2 = 5184 \]Now try \(71\): \[ 71^2 = 5041 \]Check \(70.5\): \[ 70.5^2 = 4970.25 \]\(\sqrt{5000}\) is between \(70.5\) and \(71\).

Step 3: Test Closest Decimals

Continue testing for decimals between 70.5 and 71. Try 70.6, 70.7, and 70.8.\(70.6^2 = 4984.36\)\(70.7^2 = 4998.49\)\(70.8^2 = 5012.64\)\(\sqrt{5000}\) is closest to 70.7.

Step 4: Conclusion

Since \(70.7^2\) brings us closest to 5000 without exceeding it significantly, we conclude:\[ \sqrt{5000} \approx 70.7 \] Thus, \(x = \sqrt{5000}\) is approximately 70.7 to the nearest tenth.

Key Concepts

Estimation of Square Roots

The process of estimating square roots helps us determine which two whole numbers a square root lies between. In the provided exercise, our task is to estimate the square root of 5000. We start by identifying perfect squares that are near 5000. We evaluate that 70 squared (4900) and 80 squared (6400) perfectly enclose 5000. Thus, we can conclude that \(\sqrt{5000}\) lies between 70 and 80.

Further refinement is necessary to bring us closer to the exact value. By testing squares of numbers like 71 and 72, and noting their squares (5041 and 5184 respectively), we realize that 71 squared is closer to 5000. Estimation involves a systematic approach, slowly narrowing down possibilities until reaching a satisfactory precision.

Rounding Decimals

Rounding decimals can make numbers easier to work with, especially when finding an approximate value. In this exercise, we're aiming to find \(x = \sqrt{5000}\) to the nearest tenth. When decimals are involved, it is crucial to understand rounding rules.

To round to the nearest tenth, look at the digit in the hundredths place. If it's 5 or more, round the tenths place up. If it's less than 5, the tenths place remains the same. Here, we determined the square root to approximately be 70.7 after rounding, ensuring it is close enough to the real value but easier to use in subsequent calculations.

Number Sense

Number sense is an intuitive understanding of numbers and their relationships. It plays a critical role when working with concepts like square roots. By having a good number sense, you can quickly estimate where a number such as \(\sqrt{5000}\) might lie.

Recognizing that 5000 is closer to 4900 than to 6400 informs us immediately that our answer will be just over 70. This basic understanding can guide a more detailed exploration and verification process. Developing such instincts is vital in mathematical problem-solving, making complex problems more approachable.

Prealgebra Concepts

Understanding prealgebra concepts is key to solving problems that involve estimation and approximation, like finding the square root of 5000. In this context, knowledge of squares and roots is foundational.

Developing skills in prealgebra includes recognizing and computing square numbers, understanding the relationship between numbers and their squares, and knowing how to manipulate and approximate square roots. This forms a basis for more advanced mathematical topics. Through practice and application, students build a solid arithmetic foundation, which will aid in confidently handling more complicated algebraic tasks in the future.