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

Step-by-Step Solution

Answer

The answer is 90.8.

Step 1: Expression Understanding

The expression given is \( \pi r^{2} h \), which represents the volume of a cylinder. We need to substitute the values for \( \pi, r, \) and \( h \).

Step 2: Substitute Given Values

Replace \( \pi \) with 3.14, \( r \) with 1.6, and \( h \) with 11.2. The expression becomes: \( 3.14 \times (1.6)^2 \times 11.2 \).

Step 3: Evaluate the Squared Term

Compute \( (1.6)^2 \). Using a calculator, \( (1.6)^2 = 2.56 \). Now the expression is \( 3.14 \times 2.56 \times 11.2 \).

Step 4: Multiply the Remaining Terms

First multiply \( 3.14 \times 2.56 \). It equals approximately \( 8.0384 \). Now multiply this result by 11.2.

Step 5: Calculate Final Result

Multiply \( 8.0384 \times 11.2 \) using a calculator. The result is approximately \( 90.829 \).

Step 6: Round to Nearest Tenth

Round 90.829 to the nearest tenth, which gives 90.8.

Key Concepts

Volume of a Cylinder

Understanding the volume of a cylinder can help in many real-life situations and mathematical problems. The formula for the volume of a cylinder is given by \( V = \pi r^{2} h \), where \( \pi \), \( r \), and \( h \) represent:

• \( \pi \) - Irregular constant often approximated as 3.14
• \( r \) - Radius of the circular base
• \( h \) - Height of the cylinder

To interpret the formula, think of it as finding how much space is inside a cylinder. Begin by calculating the area of the base (\( \pi r^{2} \)), which is essentially a circle, and then multiply it by the height (\( h \)).
This process results in a three-dimensional volume, measured in cubic units.
It's important to correctly identify and substitute the given values for radius and height to compute the volume accurately. Once you understand this foundational step, applying the formula becomes simple, regardless of the numeric values provided.

Calculator Usage

When working with complex expressions involving multiplication and powers, calculators become an indispensable tool to ensure accuracy and speed.Begin by inputting the expression step by step:

• Square the radius first: Input \( 1.6 \times 1.6 \) to get \( 2.56 \).
• Next, multiply \( 3.14 \times 2.56 \) to obtain \( 8.0384 \).
• Finally, multiply \( 8.0384 \times 11.2 \) to achieve \( 90.829 \).

Using a calculator minimizes the likelihood of errors that can easily occur during manual computations, especially when dealing with decimals or irrational numbers like \( \pi \).
Make sure to follow the order of operations correctly, dealing with exponents before multiplication or other operations. This practice helps achieve consistent results every time you calculate using algebraic expressions.

Rounding to the Nearest Tenth

Rounding numbers is essential for simplifying results, especially when exact values aren't necessary. This practice involves:

• Identifying the decimal place to which you want to round (the first decimal place for nearest tenth).
• Looking at the digit in the next decimal place (hundredths place) to decide if you should round up or down.

For instance, in the calculation result \( 90.829 \), you look at the hundredths digit, which is \( 2 \). Since \( 2 \) is less than \( 5 \), you leave the tenths place as it is, rounding \( 90.829 \) down to \( 90.8 \).
Understanding how to round numbers helps simplify results, making them easier to interpret and use in subsequent calculations or real-world scenarios.