Problem 54 - Algebra A Combined Function

Question

The silo shown is in the shape of a cylinder. If its radius is \(4 x\) meters and its height is \(5 x^{3}\) meters, find its volume. Do not approximate \(\pi\).

Step-by-Step Solution

Answer

The volume of the silo is \(80\pi x^5\) cubic meters.

Step 1: Understand the formula for the volume of a cylinder

The formula to calculate the volume of a cylinder is given by \[ V = \pi r^2 h \]where \(r\) is the radius of the cylinder, and \(h\) is the height of the cylinder.

Step 2: Identify the given measurements

We know that the radius \( r = 4x \) meters and the height \( h = 5x^3 \) meters.

Step 3: Plug the values into the volume formula

Substitute \( r = 4x \) and \( h = 5x^3 \) into the formula:\[ V = \pi (4x)^2 (5x^3) \]

Step 4: Simplify the expression

Calculate \((4x)^2 = 16x^2\). Now, substitute it back into the volume formula:\[ V = \pi \, \cdot \, 16x^2 \, \cdot \, 5x^3 \]

Step 5: Further simplify to find the volume

Multiply the numerical coefficients and combine the powers of \(x\):\[ V = \pi \, \cdot \, 80x^{5} \]Therefore, the volume of the cylinder is \( V = 80\pi x^5 \) cubic meters.

Key Concepts

Cylinder Geometry

A cylinder is one of the most commonly encountered 3D shapes in both geometry and real-world contexts. It features two parallel bases that are usually circular and are connected by a curved surface. Understanding the basic properties of a cylinder can help in solving various geometric problems.

Key characteristics of a cylinder include:

• The bases: Both the top and bottom surfaces of a cylinder are called its bases. They are congruent and parallel to each other.
• The height: This is the perpendicular distance between the bases. It is sometimes denoted as the vertical side that connects these bases.
• The radius: The distance from the center to the boundary of the circular base is termed as the radius.

Cylinders are prevalent in everyday life, from simple objects like cans to complex structures like silos. This real-world context not only helps make sense of volume calculations but also enhances their relatability.

Algebraic Expressions

In mathematics, algebraic expressions are combinations of constants, variables, and algebraic operations like addition and multiplication. Understanding how to handle algebraic expressions is crucial, especially when dealing with geometric formulas.

For instance, consider the silo problem where we have expressions for the radius and height as functions of a variable, in this case,

• Radius: \(4x\)
• Height: \(5x^3\)

These expressions allow us to study relationships and calculate more complex expressions such as volume. Manipulating these requires knowledge of:

• Exponents: Know how to multiply terms like \((4x)^2\), which is crucial for expanding expressions.
• Coefficients: Handle multiplying constants and variables separately before combining them.

By mastering these skills, solving for volumes or other properties becomes straightforward.

Mathematical Formulas

Formulas are the blueprints of solving mathematical problems. Especially in geometry, knowing the right formulas is key to unlocking a variety of solutions. The volume of a cylinder is one such area where a specific formula applies.

The formula for the volume of a cylinder is \( V = \pi r^2 h \), important for calculating how much space is contained inside a cylindrical object. Here's what each component represents:

• \( \pi \): This constant (approximately 3.14159) reflects the ratio of circumference to diameter for a circle.
• \( r^2 \): The square of the radius of the base, representing how large the base area is.
• \( h \): The height of the cylinder, determining its extension upward and thereby affecting the volume.

By plugging the values of the radius and height from algebraic expressions into this formula, one can compute the exact volume of any cylinder. Understanding and applying formulas simplifies and speeds up mathematical problem-solving.