Problem 61 - 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. $$ \pi r^{2} h, \quad \pi=3.14, r=4.8, \text { and } h=15.1 $$

Step-by-Step Solution

Answer

The volume is 1092.4.

Step 1: Identify the Expression to Evaluate

The algebraic expression given is the formula for the volume of a cylinder: \( \pi r^2 h \). We will use this formula to find the volume of the cylinder when \( \pi = 3.14 \), \( r = 4.8 \), and \( h = 15.1 \).

Step 2: Substitute the Values into the Formula

We substitute \( \pi = 3.14 \), \( r = 4.8 \), and \( h = 15.1 \) into the expression: \( 3.14 \times (4.8)^2 \times 15.1 \).

Step 3: Calculate \( r^2 \)

Calculate \( r^2 \) by squaring 4.8: \( 4.8 \times 4.8 = 23.04 \).

Step 4: Multiply by \( \, h \, \) and \( \, \pi \, \) to Find the Volume

Take the result from the previous step \((23.04)\) and multiply it by \( h (15.1) \): \( 23.04 \times 15.1 = 347.904 \). Then, multiply this result by \( \pi (3.14) \): \( 3.14 \times 347.904 = 1092.41 \).

Step 5: Round the Answer to the Nearest Tenth

The final step is to round 1092.41 to the nearest tenth. We observe that the first decimal place is 4, so we round down to 1092.4.

Key Concepts

Algebraic Expressions

Algebraic expressions play a key role in mathematics, especially when expressing complex calculations in a simpler form. In this exercise, we're looking at the volume of a cylinder, which is described algebraically as \( \pi r^2 h \). Here's what each part of the expression means:

• \(\pi\): A constant that represents the ratio of the circumference of a circle to its diameter, commonly approximated as 3.14.


• \(r^2\): The radius \(r\) squared, which gives the area of the base of the cylinder.


• \(h\): The height of the cylinder.

Substituting the given values \(\pi = 3.14\), \(r = 4.8\), and \(h = 15.1\) into our algebraic expression allows us to calculate the cylinder's volume. The whole purpose of algebraic expressions is to provide a general method for solving problems with potentially different numbers which makes them a powerful tool in mathematics.

Using Calculators

Calculators are incredibly helpful when evaluating algebraic expressions, especially when dealing with large numbers or multiple operations, as in our cylinder volume problem. They assist in ensuring accuracy and saving time. To use a calculator effectively, ensure the correct order of operations:

• Enter parentheses first: If needed, enclosed calculations should be made first, such as the \(r^2\) here.


• Follow the sequence: After parentheses, perform squares, followed by multiplications and other operations.


• Double-check: Always look over your input to ensure it matches the algebraic expression you're calculating.

In this problem, a calculator helps us first square \(4.8\) and then multiply the result by the height and \(\pi\). This not only speeds up the process but helps prevent errors from manual calculations. The device then also easily handles the rounding step, giving us the precise value to the nearest tenth, simplifying our work immensely.

Mathematical Formulas

Mathematical formulas, like the one we examine here for the volume of a cylinder, provide structured ways to tackle calculations that would otherwise be complicated to manage. A formula in mathematics acts like a blueprint:

• Standardization: Formulas give a structured approach to problem-solving, like \(\pi r^2 h\), designed to find the cylinder's volume for any given values of \(r\) and \(h\).


• Universality: With set formulas, you can calculate results across different disciplines and scenarios.


• Accuracy: A formula ensures you account for each variable correctly regarding what is necessary to solve the problem.

In essence, mathematical formulas harness the power of algebra to translate complex concepts into solvable equations. Learning to identify and apply the right formula, like with the cylinder's volume, aids greatly in problem-solving and mathematical understanding, making calculations much more manageable.