Problem 92 - Algebra 1: Concepts and Skills

Question

Which two consecutive integers does \(\sqrt{200}\) fall between?? a. 10 and 11 b. 13 and 14 c. 14 and 15 d. 19 and 20

Step-by-Step Solution

Answer

The square root of 200 falls between the integers 14 and 15, so the correct answer is c. 14 and 15.

Step 1: Finding the Square Root of 200

The first step is to calculate the square root of 200. To do this, a scientific calculator can be used, or square root property can be applied. By calculation, \(\sqrt{200} \approx 14.1421\)

Step 2: Analyzing the Results

Once we have the square root of 200 which is approximately 14.1421, we look at this number to decide between which two consecutive integers it falls.

Step 3: Determine the Integers

The number 14.1421 is clearly greater than 14 but less than 15. Hence, it falls between the integers 14 and 15.

Key Concepts

Understanding Consecutive Integers

In mathematics, consecutive integers are numbers that follow each other in the natural order without any gaps. For example, 1, 2, 3, and so on. This sequence pattern helps in identifying intervals between values like square roots. Knowing consecutive integers is essential when estimating where a square root falls between two whole numbers. In the exercise we have, we need to find which two integers sandwich the value of \(\sqrt{200}\) closely.

Concept of Approximation

Approximation is a mathematical technique to find a number that is close enough to the actual value for a given purpose. This is especially useful when dealing with irrational numbers like square roots, which can be never-ending decimals. In the exercise, the square root of 200 is approximately 14.1421, which means it isn’t exact but very close. Approximating helps you make quick decisions about where certain calculations land in respect to whole numbers or fractions. This concept can make solving math problems less intimidating by providing a simplified perspective.

Square Root Calculation Basics

A square root of a number is a value that, when multiplied by itself, gives the original number back. Calculating a square root depends on whether the number is a perfect square or not. A perfect square is an integer that is the square of another integer. For instance, 16 is a perfect square because it is 4 squared. For numbers like 200, which are not perfect squares, calculators are often used for quick precision. In manual calculation, estimating between perfect squares such as 196 \(14^2\) and 225 \(15^2\) helps locate the position of \(\sqrt{200}\) between them.

Mathematical Estimation Techniques

Estimation techniques in mathematics provide strategic ways to gauge values without needing exact numbers. Techniques can include:

• Rounding numbers to the nearest integer for easier computation.
• Using known benchmark values for quicker insights, such as knowing squares of numbers close to the target.
• Breaking down problems into smaller parts can simplify comprehension or calculation.

In our example, understanding that \(14^2 = 196\) and \(15^2 = 225\) allows us to estimate that \(\sqrt{200}\) rests between 14 and 15. Estimation helps validate results quickly, particularly useful in exams or time-sensitive calculations.