Problem 62 - 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. \(2 \pi r^{2}+2 \pi r h, \pi=3.14, r=3.9\), and \(h=17.6\)

Step-by-Step Solution

Answer

The expression evaluates to 526.4.

Step 1: Identify the Expression Components

Identify the components of the algebraic expression. The expression given is \(2 \pi r^2 + 2 \pi r h\), where \(\pi = 3.14\), \(r = 3.9\), and \(h = 17.6\). The terms are \(2 \pi r^2\) and \(2 \pi r h\).

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

First, calculate \(r^2\) as \(r = 3.9\). So, \(r^2 = 3.9^2 = 15.21\). Then multiply by \(2\) and \(\pi\): \(2 \times 3.14 \times 15.21 = 95.5868\).

Step 3: Calculate \(2 \pi r h\)

Multiply \(r\) and \(h\) first: \(3.9 \times 17.6 = 68.64\). Then, multiply by \(2\) and \(\pi\): \(2 \times 3.14 \times 68.64 = 430.7904\).

Step 4: Add the Results

Add the numbers calculated in Step 2 and Step 3: \(95.5868 + 430.7904 = 526.3772\).

Step 5: Round to the Nearest Tenth

Round the result \(526.3772\) to the nearest tenth: \(526.4\).

Key Concepts

Evaluating Expressions

Evaluating algebraic expressions is a fundamental skill in algebra that allows you to calculate specific values based on given variables. In the exercise, you're asked to evaluate the expression \(2 \pi r^2 + 2 \pi r h\) using the values \(\pi = 3.14\), \(r = 3.9\), and \(h = 17.6\).
To evaluate, follow these steps:

• Substitute the known values into the expression. This transforms the expression from an abstract formula into a numerical equation you can solve. Here, substitute to get: \(2 \times 3.14 \times 3.9^2 + 2 \times 3.14 \times 3.9 \times 17.6\).
• Perform the arithmetic operations. These include squaring \(r\), multiplication, and addition. It is crucial to follow the order of operations (PEMDAS/BODMAS) which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

The process of evaluating expressions can be practiced with various values and different algebraic expressions to gain confidence in solving these types of mathematical problems.

Rounding to the Nearest Tenth

Rounding numbers make them easier to work with, especially when dealing with long decimals. Rounding to the nearest tenth involves finding the first digit after the decimal point and adjusting it based on the digit that follows.
Here's how to do it:

• Identify the tenths place. In the number \(526.3772\), this is the digit \(3\) immediately after the decimal.
• Look at the digit in the hundredths place to decide. If it is 5 or greater, you round up. If it is less than 5, you let the tenths digit remain as is. Here, \(7\) is more than \(5\), so \(3\) becomes \(4\).
• The rounded number becomes \(526.4\).

Rounding helps simplify complex numbers and is particularly useful in making approximations which are easier for communication and understanding.

Using a Calculator

Using a calculator is a common practice to aid in solving complex and lengthy calculations accurately and efficiently. Understanding how to use this tool effectively is essential for evaluating algebraic expressions like the ones in this exercise.
Here’s a simple guide to using a calculator efficiently:

• Enter numbers carefully: Ensure you enter each value correctly by double-checking digits as you type.
• Use parentheses to control operation priority. Calculators follow the order of operations, so using parentheses can force calculations to happen in the desired order.
• Double-check the result: Since human error in entering data is common, confirming the result by recalculating can ensure accuracy.

By mastering the use of calculators, you save time and minimize errors, which is particularly helpful when dealing with multiple steps and larger numbers in expressions.