Problem 63 - 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. \(2 \pi r^{2}+2 \pi r h, \quad \pi=3.14, r=7.8\), and \(h=21.2\)

Step-by-Step Solution

Answer

The evaluated expression is approximately 1420.3.

Step 1: Understand the Expression

The expression provided is a formula for the surface area of a cylinder: \(2 \pi r^2 + 2 \pi r h\). Here, \(r\) is the radius of the base of the cylinder, and \(h\) is the height of the cylinder. Our task is to evaluate this expression using the given values for \(\pi\), \(r\), and \(h\).

Step 2: Substitute the Given Values

We substitute the given values into the expression: \(\pi = 3.14\), \(r = 7.8\), and \(h = 21.2\). The expression becomes: \[ 2 \times 3.14 \times (7.8)^2 + 2 \times 3.14 \times 7.8 \times 21.2 \]

Step 3: Calculate Each Component - Circular Base Area

First, calculate the area of the two circular bases, which is \(2 \times 3.14 \times (7.8)^2\). Compute \((7.8)^2 = 60.84\), and then the expression is:\[ 2 \times 3.14 \times 60.84 = 381.6816 \]

Step 4: Calculate Each Component - Lateral Surface Area

Now, calculate the lateral surface area, which is \(2 \times 3.14 \times 7.8 \times 21.2\). Perform the multiplication step-by-step: 1. \(7.8 \times 21.2 = 165.36\)2. \(2 \times 3.14 = 6.28\)3. \(6.28 \times 165.36 = 1038.61\)

Step 5: Add the Calculated Areas

Finally, add the two calculated components together to find the total surface area:\[ 381.6816 + 1038.6012 = 1420.2828 \]

Step 6: Round to the Nearest Tenth

Round the final answer to the nearest tenth. The number \(1420.2828\) rounds to \(1420.3\) when rounded to the nearest tenth.

Key Concepts

Surface Area of a Cylinder

The surface area of a cylinder is an important concept in geometry, especially in solving real-world problems that involve cylindrical shapes. A cylinder consists of two circular bases and a curved surface, often referred to as the lateral area. To find the total surface area, you need to calculate the area of both bases and the lateral surface area.

The formula to find the surface area of a cylinder is given by:

• Surface Area = \(2 \pi r^2 + 2 \pi r h\)

Here:

• \(\pi\) is a mathematical constant approximately equal to 3.14.
• \(r\) is the radius of the circular base.
• \(h\) is the height of the cylinder.

To visualize this, imagine unwrapping a cylindrical can. The top and bottom are the circles, and the curved side becomes a rectangle when laid flat.

Understanding this structure helps you in calculating surface area effectively. Simply find the area of the two circular ends and add it to the rectangle's area, which measures the side around the cylinder.

Substitution in Expressions

Substitution is a fundamental skill in algebra where you replace variables in an expression with their given numeric values. This process allows you to solve problems with specific data points, making the problem easier to relate to and understand.

To substitute, carefully plug in the values into the given algebraic formula or expression. For example, in the expression for the surface area of a cylinder, we have:

• The original expression: \(2 \pi r^2 + 2 \pi r h\)
• Given: \(\pi = 3.14\), \(r = 7.8\), and \(h = 21.2\)

Perform substitution like so:

• Replace \(\pi\) with 3.14, \(r\) with 7.8, and \(h\) with 21.2.
• This generates the new numerical expression: \ \(2 \times 3.14 \times (7.8)^2 + 2 \times 3.14 \times 7.8 \times 21.2\)

Substitution helps you see how each part of the formula contributes to the final result. Make sure your calculations are accurate at each step to achieve a precise solution.

Rounding to the Nearest Tenth

Rounding is a useful mathematical technique aimed at simplifying numbers, making them easier to work with while maintaining a fair sense of accuracy. When rounding to the nearest tenth, you focus on the first digit to the right of the decimal point.

Here's how to round a number to the nearest tenth:

• Identify the digit in the tenths place (the first digit after the decimal point).
• Look at the next digit (the hundredths spot). This determines whether to round the tenths digit up or leave it unchanged.
• If the hundredths digit is 5 or greater, increase the tenths digit by 1.
• If it is less than 5, keep the tenths digit as is.

For example, when rounding 1420.2828 to the nearest tenth, observe:

• The tenths digit is 2, and the hundredths digit is 8.
• Since 8 is greater than 5, increase 2 to 3.
• The rounded number becomes 1420.3.

Rounding maintains clarity in your results by eliminating less significant digits, especially useful in lengthy calculations.