Problem 60 - 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. \(\pi r^{2} h, \quad \pi=3.14, r=1.6\), and \(h=11.2\)

Step-by-Step Solution

Answer

90.8

Step 1: Understand the Formula

The algebraic expression given is \( \pi r^2 h \), which represents the formula for the volume of a cylinder. In this formula, \( \pi \) is a constant approximately equal to 3.14, \( r \) is the radius, and \( h \) is the height.

Step 2: Substitute Values into the Expression

Substitute the given values into the expression: \( \pi = 3.14 \), \( r = 1.6 \), and \( h = 11.2 \). This gives us the expression \( 3.14 \times (1.6)^2 \times 11.2 \).

Step 3: Evaluate the Square of the Radius

Calculate \( (1.6)^2 = 1.6 \times 1.6 \). Use a calculator to evaluate this, which results in \( 2.56 \).

Step 4: Calculate the Volume

Now we have the expression \( 3.14 \times 2.56 \times 11.2 \). Use a calculator to find the product. First, calculate \( 3.14 \times 2.56 = 8.0384 \). Then, multiply by 11.2: \( 8.0384 \times 11.2 = 90.83008 \).

Step 5: Round to the Nearest Tenth

Round the final result to the nearest tenth. The number 90.83008, rounded to the nearest tenth, is 90.8.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition, subtraction, multiplication, and division) that represent a value. In the given exercise, we focus on the algebraic expression for calculating the volume of a cylinder, \[ \pi r^2 h \] Understanding this expression is crucial. Here, \( \pi \) is a well-known mathematical constant, approximately equal to 3.14, which helps in calculations involving circles. The variable \( r \) stands for the radius of the circular base of the cylinder, and \( h \) denotes the height of the cylinder.
To fully comprehend algebraic expressions, practice breaking down each part. This would help you know what every symbol stands for and how they work together to give a mathematical description of a scenario or geometric shape.

Substitution in Formulas

Substitution is a key technique in algebra where you replace variables with numbers to solve equations or evaluate expressions. In this exercise, you substitute the given values into the formula for the volume of a cylinder. The formula \[ \pi r^2 h \] becomes \[ 3.14 \times (1.6)^2 \times 11.2 \] when you insert \( \pi = 3.14 \), \( r = 1.6 \), and \( h = 11.2 \).
This step is crucial to solving the problem as it turns abstract mathematical expressions into something concrete you can calculate. It also helps ensure that everyone using the formula applies it in the same way, leading to consistent results across different scenarios.
When substituting, always double-check that you have inserted the correct values in their correct spots to avoid errors.

Decimal Rounding

Decimal rounding is the process of approximating a number to make it simpler, yet keeping it close to its original value. It's especially useful in measurements and calculations to make results easier to read and use. In this exercise, after calculating the volume of the cylinder, we ended up with 90.83008.
To round this number to the nearest tenth, you need to look at the first digit after the decimal point. If this digit is 5 or higher, you round up. If it's 4 or lower, you round down. So, 90.83008 becomes 90.8. This technique is important to ensure that results are precise enough for practical use without unnecessary detail. Mastering rounding can save time and helps in achieving accurate results in exams as well.

Calculator Usage in Math

Using a calculator is a helpful skill in math. It allows you to efficiently perform calculations that would be tedious by hand, such as multiplying decimals or computing squares. In this exercise, the calculator was key to evaluating \[(1.6)^2\] and then multiplying it by 3.14 and 11.2.
Calculators can quickly compute these operations but understanding the steps behind the calculations ensures you’re using it effectively. Always make sure your calculator is set to the correct mode (degree/radian, decimal/fraction) for the type of problem.
Additionally, become familiar with the keys on your calculator to speed up your work and to ensure precision. It's a useful tool for verifying your manual calculations and can help in understanding complex problems.