Problem 19 - Elementary Algebra

Question

Find the volume of a cylinder with a diameter of 15 in. and a height of \(24 \mathrm{in}\).

Step-by-Step Solution

Answer

The volume of the cylinder is 4240.5 cubic inches.

Step 1 - Identify the formula for the volume of a cylinder

The formula to find the volume of a cylinder is \[ V = \pi r^2 h \] where \( V \) is the volume, \( \pi \) is a constant (approximately 3.14), \( r \) is the radius, and \( h \) is the height.

Step 2 - Determine the radius of the cylinder

The diameter of the cylinder is given as 15 inches. The radius is half of the diameter. Therefore, \[ r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ inches} \]

Step 3 - Plug in the values to the volume formula

Now, we can substitute the radius \( r = 7.5 \text{ inches} \) and the height \( h = 24 \text{ inches} \) into the formula: \[ V = \pi (7.5)^2 (24) \]

Step 4 - Calculate the volume

First, calculate the square of the radius: \[ (7.5)^2 = 56.25 \] Then, multiply by the height: \[ 56.25 \times 24 = 1350 \] Finally, multiply by \( \pi \) (approximately 3.14): \[ V = 1350 \times 3.14 = 4240.5 \text{ cubic inches} \]

Key Concepts

Cylinder Volume Formula

To find the volume of a cylinder, we use a specific formula. This formula is crucial in geometry and is quite simple to remember. The formula for the volume of a cylinder is given by


substantiate substantiate substantiate Insert the radius value and height into the formula, then calculate to find the volume.

Radius Calculation

Determining your cylinder's radius is an essential step. Given the diameter, you can easily find the radius. Remember, the radius is half of the diameter.

For example:

• A cylinder has a diameter of 15 inches.
• The radius will be half of this diameter, which is calculated as follows:
  
  Understanding and calculating the radius provides the foundation for finding the cylinder's total volume.

Geometry in Algebra

Geometry often intersects with algebra, especially in problems like finding the volume of a cylinder. To blend these two fields, we need to recognize shapes, understand their properties, and apply algebraic formulas.

Here’s how these concepts combine:

• Identify the shape (in this case, a cylinder).
• Recall the volume formula involving algebraic terms.
• Substitute the geometric measurements into the algebraic formula.

Both algebra and geometry contribute to a comprehensive understanding and solving of the problem effectively.

Unit Conversion

Paying attention to units is critical when calculating volume. Always ensure that measurements are in the same units.

Here's a quick guide:

• Height and diameter/radius must be in the same unit (e.g., inches in our case).
• Volume will be in cubic units of the chosen measurement system (e.g., cubic inches).

Consistency in units guarantees the accuracy of your final answer.