Problem 58 - 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. $$ \pi r^{2}, \quad \pi=3.14 \text { and } r=2.1 $$

Step-by-Step Solution

Answer

13.8

Step 1: Substitute Values into the Expression

First, substitute the given values into the algebraic expression \(\pi r^2\). Replace \(\pi\) with 3.14 and \(r\) with 2.1. The expression becomes: \(3.14 \times (2.1)^2\).

Step 2: Calculate the Square of the Radius

Now, calculate \((2.1)^2\). This means multiplying 2.1 by itself: \(2.1 \times 2.1 = 4.41\).

Step 3: Multiply by Pi

Next, multiply the square of the radius by \(\pi\). So, multiply 3.14 by 4.41: \(3.14 \times 4.41\).

Step 4: Use Calculator and Round the Answer

Use a calculator to find \(3.14 \times 4.41 = 13.8474\). Round this result to the nearest tenth to get 13.8.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra, where you replace variables in an expression with given numerical values. This technique is especially useful in evaluating expressions like \(\pi r^2\). It involves the following steps:

• Identify the variables in the expression. For \(\pi r^2\), \pi\ and \r\ are the variables.
• Substitute each variable with its given value. In this exercise, replace \pi\ with 3.14 and \r\ with 2.1. This changes the expression to \(3.14 \times (2.1)^2\).

Remember, proper substitution allows for accurate calculations, making it a crucial skill when working with algebraic formulas.

Calculating Squares

Calculating squares is all about multiplying a number by itself. When the problem gives you \(r = 2.1\), and asks for \(r^2\), you need to perform the operation:

• Multiply the radius (2.1) by itself: \(2.1 \times 2.1 = 4.41\).

Understanding squares is essential because it appears frequently in algebra, especially in formulas involving areas, like \[\pi r^2\], which is used to find the area of a circle. The square of a number gives you the area of a square with sides equal to that number, which helps in visualizing the magnitude of squares.
Being comfortable with squaring numbers helps simplify many algebraic expressions and is a stepping stone to more complex calculations.

Rounding Numbers

Rounding is a method of reducing the digits in a number while keeping its value close to the original. This process makes calculations simpler and results easier to communicate. To round a number to the nearest tenth, follow these steps:

• Identify the hundredths place. For the number 13.8474, 4 is in the hundredths place.
• If the hundredths digit is 5 or greater, round the tenths digit up. Otherwise, keep the tenths digit the same.

In our example, since 4 is less than 5, we keep the tenths digit as it is. Thus, 13.8474 rounds to 13.8.
Rounding values is useful in making numerical answers more manageable and is a standard step in presenting numerical results, ensuring your calculations maintain clarity and precision.