Problem 59 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The area is 221.6 square units.

Step 1: Understand the formula

The formula given is the area of a circle, \( \pi r^2 \). Here, \( \pi \) is a constant approximately equal to 3.14, and \( r \) is the radius of the circle, which is given as 8.4.

Step 2: Substitute the values

Replace \( \pi \) with 3.14 and \( r \) with 8.4 in the formula \( \pi r^2 \). The expression becomes \( 3.14 \times (8.4)^2 \).

Step 3: Calculate \( r^2 \)

First, square the radius: \( r^2 = (8.4)^2 = 8.4 \times 8.4 = 70.56 \).

Step 4: Calculate \( \pi r^2 \)

Multiply the squared radius by \( \pi \): \( 3.14 \times 70.56 \). Use a calculator to perform the multiplication, which gives \( 221.5584 \).

Step 5: Round to the nearest tenth

Round the result from Step 4 to the nearest tenth: 221.5584 rounds to 221.6.

Key Concepts

Algebraic Expressions

Algebraic expressions are like recipes for mathematics. They help us calculate specific values by combining constants, variables, and mathematical operations. In our exercise, the algebraic expression is the formula for the area of a circle: \( \pi r^2 \). This expression requires specific input values, in this case, the constant \( \pi \) and the variable \( r \) for radius.

Here’s how we use algebraic expressions in this context:

• Identify the formula: Understand the components involved, like constants and variables.
• Substitute the numbers: Replace the variables with given values.
• Simplify: Calculate the result according to mathematical rules.

By following these steps, you transform the algebraic expression into a concrete numerical answer, which is often presented as the area, volume, or another measurement. It's a powerful way to connect abstract formulas to tangible outcomes.

Radius

The radius is a crucial measurement that defines every circle. It's the distance from the center of the circle to any point on its circumference. In practical exercises, we are often given the radius as a starting value, like in our problem, where \( r = 8.4 \).

Understanding the radius is vital for several reasons:

• Proportional Significance: The radius is directly proportional to the size of the circle; a larger radius means a bigger circle.
• Squared Radius: In formulas like the area calculation \( (\pi r^2) \), the radius is squared. This means its impact is greater than just a linear measurement.
• Units Matter: The units you use for the radius will affect the final measurement of the area (e.g., meters, centimeters).

When calculating a circle's area, knowing the radius allows you to move smoothly to the next step of squaring it and then using it in further calculations.

Pi

Pi (\( \pi \)) is a unique and fascinating constant in mathematics, always appearing in calculations involving circles. It represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159.

Key aspects of \( \pi \) to understand include:

• Approximation: In many practical scenarios, \( \pi \) is approximated to 3.14 or 22/7, which simplifies our calculations.
• Constant Nature: Unlike variables, \( \pi \) never changes. Its value remains the same regardless of the circle's size.
• Significance in Formulas: It's essential in determining the circumference and area of a circle, showcasing its central role in geometry.

In calculations like our exercise, the constant \( \pi \) ensures uniformity and accuracy when determining the area of circles, making it a cornerstone in mathematical computations.