Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Approximate each expression to the nearest hundredth. $$\frac{\sqrt{\pi-1}}{\sqrt{1+\pi}}$$

Step-by-Step Solution

Answer

0.72

Step 1: Evaluate Expressions Inside the Square Roots

First, approximate the value of \( \pi \), which is approximately \( 3.14159 \). Then calculate the values inside the square roots:- \( \pi - 1 \) is approximately \( 3.14159 - 1 = 2.14159 \).- \( 1 + \pi \) is approximately \( 1 + 3.14159 = 4.14159 \).

Step 2: Calculate the Square Roots

Now calculate the square roots of the values obtained in Step 1:- \( \sqrt{2.14159} \) is approximately \( 1.46385 \).- \( \sqrt{4.14159} \) is approximately \( 2.03541 \).

Step 3: Divide the Results

Use the values from Step 2 to calculate the final division:- Divide the square root results: \( \frac{1.46385}{2.03541} \).

Step 4: Approximate the Result

Perform the division and round the result to the nearest hundredth:- \( \frac{1.46385}{2.03541} \approx 0.7192 \), which rounds to \( 0.72 \) when rounded to the nearest hundredth.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. When calculating a square root, you are essentially finding a number that, when multiplied by itself, gives the original number. In our exercise, we are calculating the square roots of the numbers that are results of expressions involving \( \pi \).
To do this efficiently, it helps to understand the meaning visually and numerically:

• Visual Understanding: Imagine a perfect square, and square root helps you to find the length of one side.
• Numerical Understanding: Recognize that the square root of a number is one of its two equal factors — for instance, \( \sqrt{4} = 2 \), because \( 2 \times 2 = 4 \).

Approximating square roots is common when the exact root is not straightforward. For example, if you needed to find \( \sqrt{2.14159} \), you have to use techniques, like estimation or calculators, to achieve a decimal approximation which is around \( 1.46385 \). This is the number we use in further calculations.

Division

Division is a basic arithmetic operation that involves splitting a number into equal parts. In our context, after finding the square roots of two values, the next step is to divide one by the other.
The division involves determining how many times the divisor fits into the dividend. To illustrate:

• Understanding Division: Think of division as sharing equally or fitting repeatedly into a starting quantity.
• Using the Example: When dividing \( 1.46385 \) by \( 2.03541 \), you are finding out how many times \( 2.03541 \) fits into \( 1.46385 \).

To approximate this, use a calculator, yielding a result of approximately \( 0.7192 \). This method is essential for breaking down problems into manageable steps and aiding in clear problem-solving processes.

Rounding

Rounding numbers is a way to simplify them to make them easier to work with while keeping their value close to the original. In mathematics, rounding is essential when a precise value is not necessary, or when limited decimal places are required for consistency or simplicity.
Here's how to approach rounding:

• Rounding to the Nearest Hundredth: Look at the third decimal place. If it is 5 or more, round the second decimal place up. If it is less than 5, keep the second decimal place as it is.
• Application: From the division in our exercise, we get approximately \( 0.7192 \). Since the third decimal place, 9, is greater than 5, we round up to \( 0.72 \).

Rounding helps focus on meaningful precision in calculations, especially in scientific, engineering, and financial fields where exact decimals aren't always necessary.

Pi Value

Pi (\( \pi \)) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it can't be exactly expressed as a simple fraction, and its decimal representation is infinite and non-repeating. The value of \( \pi \) is approximately \( 3.14159 \).
Understanding \( \pi \) is crucial because:

• Significance in Geometry: \( \pi \) is used in calculations involving circles, such as area and circumference.
• Usage in Approximation: In our exercise, we substitute \( \pi \) with its approximate value \( 3.14159 \) to make further calculations manageable.

For calculations requiring pi, using an approximate value like \( 3.14159 \) or the fraction \( \frac{22}{7} \) provides a practical approach, keeping solutions straightforward and comprehensible for simpler approximations.