Problem 59 - Intermediate Algebra

Question

Use your calculator and evaluate each of the algebraic expressions for the indicated values. Express the final answers to the nearest tenth. $$ \pi r^{2}, \quad \pi=3.14 \text { and } r=8.4 $$

Step-by-Step Solution

Answer

The evaluated expression for \( \pi r^2 \) is 221.6.

Step 1: Identify the Formula and Values

The formula given is the area of a circle: \( \pi r^2 \). We have been given \( \pi = 3.14 \) and \( r = 8.4 \).

Step 2: Substitute the Values

Substitute the given values into the formula: \( 3.14 \times (8.4)^2 \).

Step 3: Calculate the Square of Radius

Calculate \( (8.4)^2 \): \[ 8.4 \times 8.4 = 70.56 \]

Step 4: Multiply by Pi

Multiply the squared radius by \( \pi \): \[ 3.14 \times 70.56 \]

Step 5: Perform the Multiplication

Perform the multiplication: \[ 3.14 \times 70.56 = 221.5584 \]

Step 6: Round to the Nearest Tenth

Round 221.5584 to the nearest tenth, which is 221.6.

Key Concepts

Area of a Circle

When we talk about the area of a circle, we are referring to the space contained within its boundary. It's a common concept in geometry and is essential for calculating various aspects of circular shapes. The formula for finding the area is \( \pi r^2 \), where \( \pi \) is a mathematical constant approximately equal to 3.14159. In real-world problems, we often use 3.14 as an approximation.

Let's break it down:

• \( \pi \) (Pi) is a constant that relates the circumference of a circle to its diameter and is crucial in circular calculations.
• \( r \) is the radius of the circle, which is half the diameter. It defines how large the circle is.
• The formula \( \pi r^2 \) allows us to find how many square units fit inside the circle.

Knowing how to plug in the radius into this formula lets you determine the area easily. This formula is a staple in problems involving circular gardens, wheels, and other round objects.

Substitution in Formulas

Substitution is a key mathematical skill used whenever you're dealing with formulas. It involves replacing the variables in a formula with actual numbers.

For example, in the formula for the area of a circle, \( \pi r^2 \), \( \pi = 3.14 \) and \( r = 8.4 \) were given. By substituting:

• You replace \( \pi \) with 3.14.
• You replace \( r \) with 8.4.

Once this is done, the formula becomes:

• \( 3.14 \times (8.4)^2 \)

This transformation makes it possible to compute the formula result numerically instead of simply leaving it with symbolic representations. It's a highly useful process in algebra, providing concrete answers from abstract equations.

Rounding Numbers

Rounding numbers is a technique used to simplify calculations and express numbers in a more understandable form. When dealing with decimals, it's common to round to the nearest tenth, hundredth, or other decimal places.

Here's how rounding works:

• Look at the digit to the right of the place you are rounding to.
• If it is 5 or greater, increase the rounding place by one.
• If it is less than 5, leave the rounding place as it is.

In our example, after calculating \( 3.14 \times 70.56 = 221.5584 \), we need to round to the nearest tenth. The digit in the tenths place is 5, and the hundredths place is 8, which is more than 5. Therefore, we increase the tenths digit from 5 to 6.

So, 221.5584 becomes 221.6 when rounded to the nearest tenth. Rounding is a handy skill for keeping numbers manageable and communicating them more effectively.