Problem 29 - Elementary Algebra

Question

The inner diameter of a cylindrical waste basket is 9.75 in. and its height is 14.75 in. Find its volume. Round to the nearest tenth.

Step-by-Step Solution

Answer

The volume of the cylindrical waste basket is approximately 1101.4 cubic inches.

Step 1 – Identify the Formula for Volume of a Cylinder

The volume of a cylinder can be calculated using the formula: \[ V = \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.

Step 2 – Find the Radius

The inner diameter of the waste basket is given as 9.75 inches. The radius is half of the diameter: \[ r = \frac{9.75}{2} = 4.875 \text{ inches} \]

Step 3 – Calculate the Volume

Substitute the radius (4.875 inches) and height (14.75 inches) into the volume formula: \[ V = \pi (4.875)^2 (14.75) \] Calculate the square of the radius: \[ (4.875)^2 = 23.765625 \] Now multiply by the height and \( \pi \): \[ V = \pi \times 23.765625 \times 14.75 \]

Step 4 – Simplify the Expression

First, multiply 23.765625 by 14.75: \[ 23.765625 \times 14.75 = 350.53625 \] Now, multiply by \( \pi \) (approximated as 3.14159): \[ V = 3.14159 \times 350.53625 \approx 1101.4 \]

Step 5 – Round the Volume

Round the calculated volume to the nearest tenth: \[ V \approx 1101.4 \text{ cubic inches} \]

Key Concepts

Cylindrical Volume Calculation

To find the volume of a cylinder, we use the formula: equation where \(()\) is the volume, \(()\) is the radius, and \(()\) is the height. This formula helps us understand how the volume is calculated based on the dimensions of the cylinder.

When working on practical problems like this one, it's useful to follow a systematic approach:

• Identify the formula
• Determine the radius and height of the cylinder
• Substitute these values into the formula
• Simplify the resulting expression
• Round off the final result to the required precision

By following these steps, you can effectively calculate the volume of any cylindrical object.

Geometric Formulas

Geometric formulas are mathematical equations used to calculate different properties of geometrical shapes, like area, perimeter, and volume.

In the case of cylinders, the key formula to remember is: equation This formula originates from the area of the circular base (equation)), which is then multiplied by the height (equation)) of the cylinder.

Understanding where these formulas come from helps you grasp how geometry ties into real-world applications, like finding the volume of a waste basket in this exercise.

Other important geometric formulas you might encounter include:

• Area of a rectangle:equation))
• Circumference of a circle:equation))
• Surface area of a cylinder:equation))

Mastering these basic formulas will make solving geometric problems much easier.

Radius and Diameter

The radius and diameter are two key measurements of a circle.

The diameter is the distance across the circle through its center, while the radius is half of that distance.

For a cylinder, these concepts apply to its circular base.

Given the diameter, you can find the radius by dividing it by two. For example, if the diameter of the waste basket is 9.75 inches, the radius would be: equationUnderstanding these relationships is crucial for correctly using the geometric formulas.

Remember:

• Diameter = 2 x Radius
• Radius = Diameter ÷ 2

Knowing this conversion allows you to switch between radius and diameter depending on what the problem requires.

Rounding Numbers

Rounding numbers is an essential skill in mathematics and everyday life. In this exercise, we are asked to round the volume to the nearest tenth.

Here's a quick guide to rounding:

• Identify the place value you are rounding to (in this case, the tenths place)
• Look at the digit immediately to the right of this place
• If that digit is 5 or greater, round up; if it’s less than 5, round down

In the provided solution, after calculating the volume as approximately 1101.4 cubic inches, we check the digit in the hundredths place. Since it's a digit not affecting the rounding in this case, we keep it as 1101.4. This simplifies our answer and makes it easier to interpret.

Practicing rounding helps make your calculations more user-friendly and suitable for real-world applications.