Problem 13 - The Calculus with Analytic Geometry

Question

\(\sqrt{37.5}\)

Step-by-Step Solution

Answer

6.12

Step 1: Estimate the Square Root

Find two perfect squares between which 37.5 lies. \ Consider that \(36 < 37.5 < 49\). Therefore, \(6 < \sqrt{37.5} < 7\).

Step 2: Refine the Estimate

Use an approximation method to hone in on the square root. Try guessing a decimal value between 6 and 7. \ For instance, evaluate if \(6.1^2 < 37.5 \) or if \(6.2^2 > 37.5 \)

Step 3: Narrow Down Further

Calculate \(6.1^2 = 37.21\) and \(6.2^2 = 38.44\), thus confirming \(6.1 < \sqrt{37.5} < 6.2\).

Step 4: Use a Calculator for Precision

Use a calculator for a more precise value. Using a calculator, we find \( \sqrt{37.5} \approx 6.12\).

Key Concepts

Estimating Square Roots

Estimating square roots is useful when you don't need an exact number. It helps simplify math without a calculator. Start by finding two perfect squares the number lies between. If you're estimating \( \sqrt{37.5} \), note that 36 and 49 are perfect squares close to 37.5. Therefore, \( 6 < \sqrt{37.5} < 7 \). The goal is to narrow this range using basic comparisons and refinements.

Perfect Squares

Perfect squares are numbers like 1, 4, 9, 16, etc., that are squares of whole numbers. They help in estimating square roots: for example, 6 squared (\( 6^2 = 36 \)) and 7 squared (\( 7^2 = 49 \)) frame 37.5. This tells us \( \sqrt{37.5} \) is between 6 and 7. Understanding perfect squares simplifies these estimates and ensures accuracy without a calculator.

Approximation Methods

Approximation narrows down square root estimates. First, pick a decimal between your previous whole numbers. For instance, if \( 6 < \sqrt{37.5} < 7 \), try 6.1. Calculate its square (\( 6.1^2 = 37.21 \)), then 6.2 (\( 6.2^2 = 38.44 \)). If \( 6.1 < \sqrt{37.5} < 6.2 \), repeat until your estimate is satisfactory. Approximations aren't exact but are useful for a closer answer.

Calculator Usage

When precision is necessary, a calculator is essential. After estimating, use a calculator to find an accurate square root. For \( \sqrt{37.5} \, \) calculator reveals \( \sqrt{37.5} \approx 6.12 \). Start with estimations to understand the concept and confirm with a calculator for accuracy. Calculators save time and ensure exact solutions, combining estimation for deeper learning.