Problem 23 - Algebra A Combined Function

Question

Use a calculator to approximate each square root to three decimal places. Check to see that each approximation is reasonable. $$ \sqrt{200} $$

Step-by-Step Solution

Answer

The approximation of \( \sqrt{200} \) is 14.142.

Step 1: Understand the Problem

We need to approximate the square root of 200 to three decimal places using a calculator. First, understand that finding the square root of a number implies determining a value that, when multiplied by itself, gives the original number.

Step 2: Use the Calculator

Enter 200 into your calculator and use the square root function (often marked as \( \sqrt{x} \) or by pressing a specific button for square root). This should give you an initial decimal value.

Step 3: Record the Approximation

The calculator will display approximately 14.1421. To round this to three decimal places, observe the fourth decimal place: if it's 5 or higher, round up the third place; otherwise, keep the third place as it is.

Step 4: Check Reasonableness

Estimate the square root by recognizing that 196 (which is \(14^2\)) is close to 200, so \( \sqrt{200} \) should be a bit more than 14. Our approximation is 14.142, which seems reasonable.

Key Concepts

Using a Calculator

When you need to approximate square roots, a calculator can be very helpful and fast! Most calculators have a specific function for square roots, typically represented by the symbol \( \sqrt{x} \) or a dedicated button. Begin by entering the number you wish to find the square root of, which in this case is 200.

After entering 200, press the square root button to see the result. This will give you a decimal approximation instantly. Calculators are precise tools and tend to give as many decimal places as possible. However, you'll need to round the answer to the specific number of decimal places required by your problem.

Decimal Places

Understanding decimal places is crucial when rounding your square root approximation. Each number factor after the decimal point is considered a decimal place. For instance, in the number 14.1421, the digits 1, 4, 2, and 1 are in the first, second, third, and fourth decimal places, respectively.

The task may ask you to round to three decimal places. To do this, look at the fourth decimal place. If this number is 5 or higher, add one to the third decimal place. If it is lower than 5, leave the third decimal place unchanged. For \( \sqrt{200} \), the fourth decimal digit is 1, so the rounded value remains 14.142.

Mathematical Reasoning

Mathematical reasoning helps ensure your calculated approximation is sensible. Before using your calculator, think about numbers close to 200 with known square roots. For instance, you know that 14 squared is 196. This suggests that \( \sqrt{200} \) should be slightly more than 14.

After using a calculator, check if your results align with your reasoning. Does 14.142 make sense given 14 squared is 196? In this case, yes! Mathematical reasoning serves as a sanity check to confirm that your approximation seems accurate.

Estimation Techniques

Before jumping into calculation, estimation helps form a mental image of the expected result. If you know specific perfect squares, like 196 being \(14^2\), you can quickly estimate that \( \sqrt{200} \) should be a bit more than 14.

This method of estimation allows you to cross-check your answer and make sure your calculator is functioning properly. Estimation also teaches you independence from gadgets, enhancing your proficiency in mathematics. Recognizing that \( \sqrt{225} \) equals 15 further reassures you that 14.142 for \( \sqrt{200} \) is logical.