Problem 16 - Elementary Algebra

Question

Find the volume of a cylinder with a radius of \(8 \mathrm{in}\). and height of 15 in.

Step-by-Step Solution

Answer

The volume is approximately 3014.4 cubic inches.

Step 1 - Recall the Volume Formula for a Cylinder

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

Step 2 - Substitute the Given Values

We know the radius \( r \) is 8 inches and the height \( h \) is 15 inches. Substitute these values into the formula: \[ V = \pi (8)^2 (15) \]

Step 3 - Calculate the Radius Squared

First, calculate \( 8^2 \): \[ 8^2 = 64 \] So the formula now looks like: \[ V = \pi (64) (15) \]

Step 4 - Multiply by the Height

Next, multiply by the height 15: \[ 64 \times 15 = 960 \] So the formula now looks like: \[ V = \pi (960) \]

Step 5 - Complete the Calculation

Now, multiply by \( \pi \): \[ V = 960 \pi \] Since \( \pi \approx 3.14 \), \[ V \approx 960 \times 3.14 = 3014.4 \] Thus, the volume is approximately 3014.4 cubic inches.

Key Concepts

Volume of a Cylinder

To calculate the volume of a cylinder, it's essential to understand what volume means. Volume refers to the amount of space within a 3-dimensional object. For a cylinder, which is a 3D shape with two parallel circular bases and a curved surface, we use a specific mathematical formula to find its volume. This formula is the product of the area of its base and its height: \[ V = \pi r^2 h \] Here, \( V \) represents the volume, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder. The formula works because it essentially stacks the circular base area \( \pi r^2 \) repeatedly until it reaches the cylinder's height.

Geometry

Geometry is the branch of mathematics that studies the properties and relationships of points, lines, surfaces, and solids. Understanding shapes and their characteristics is crucial for problem-solving in geometry. In this case, we focus on the cylinder. A cylinder is characterized by:

• Two parallel circular bases
• A defined radius \( r \) from the center to the edge of each base
• A height \( h \) which is the perpendicular distance between the bases

Knowing these geometric properties helps us use the correct formula when calculating properties such as volume.

Mathematical Formulas

Mathematical formulas are at the heart of problem-solving in math. They provide a way to relate different quantities and simplify complex operations. For the volume of a cylinder, the formula is: \[ V = \pi r^2 h \] Using formulas correctly involves understanding each component:

• \( \pi \) (Pi) is a constant approximately equal to 3.14
• \( r \) is the radius of the cylinder's base
• \( h \) is the height of the cylinder

Substituting the values into the formula properly ensures accurate calculations. For example:

Given \( r = 8 \) inches and \( h = 15 \) inches, we substitute these values to get: \[ V = \pi (8)^2 (15) = \pi (64) (15) = \pi (960) = 960 \pi \] Finally, multiplying by Pi gives us the result in cubic inches.

Problem-Solving Steps

Solving mathematical problems, like finding the volume of a cylinder, can be broken down into manageable steps:

Recall the formula and understand what each variable represents.
Substitute the given values into the formula.
Perform the necessary mathematical operations, such as squaring the radius and multiplying by the height.
Calculate the final product after multiplying by Pi.

Following these steps ensures clarity and accuracy in your solution. For instance, substituting \( r = 8 \) and \( h = 15 \) into \( V = \pi r^2 h \), we calculate:
\[ V = \pi (8)^2 (15) = \pi (64) (15) = \pi (960) = 960 \pi = 3014.4 \] cubic inches. This step-by-step approach helps break down the problem into simple, easy-to-follow parts.