Problem 3 - Calculus Volume 3

Question

The volume of a right circular cylinder is calculated by a function of two variables, \(V(x, y)=\pi x^{2} y,\) where \(x\) is the radius of the right circular cylinder and \(y\) represents the height of the cylinder. Evaluate \(V(2,5)\) and explain what this means.

Step-by-Step Solution

Answer

The volume of the cylinder is approximately 62.83 cubic units.

Step 1: Substitute Values for Variables

To evaluate the function \( V(x, y) = \pi x^2 y \) for specific values, begin by substituting \( x = 2 \) and \( y = 5 \) into the equation. This results in the expression \( V(2, 5) = \pi (2)^2 (5) \).

Step 2: Simplify the Expression

Next, simplify the expression \( \pi (2)^2 (5) \). Calculate \( (2)^2 \) which is 4, then multiply by 5 to get 20. The expression becomes \( V(2,5) = \pi imes 20 \).

Step 3: Calculate the Volume

Multiply \( 20 \) by \( \pi \) (approximately 3.14159) to find the volume. This gives \( V(2, 5) = 62.8318 \) cubic units (using the approximation of \( \pi \)).

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that deals with functions of more than one variable. It explores how these functions behave and change in space. In the exercise we're looking at, the volume of the cylinder depends on two variables: the radius (\(x\)) and the height (\(y\)). This is what makes it a multivariable function:

• The function \(V(x, y) = \pi x^2 y\) shows how volume changes with radius and height.
• Each variable contributes differently to the value of the function.

Multivariable calculus is important because it allows us to analyze and predict outcomes in systems that depend on multiple factors. Understanding how to work with such functions is key for solving real-world engineering and physics problems.

Cylinder Volume

The concept of the cylinder volume is vital in understanding how volumes of 3D objects are calculated. A cylinder consists of two parallel circular bases and a curved surface that connects these bases:

• The formula \(V = \pi r^2 h\) derives from basic geometric principles where \(r\) is the radius and \(h\) is the height of the cylinder.
• This means any change in the radius or the height directly affects the volume.

The cylinder volume formula is a classic example of a multivariable function because it involves more than one independent variable. Calculating this in real-world scenarios helps in tasks like finding the amount of water a cylindrical tank can hold or the volume of material needed to create a cylindrical object.

Function Evaluation

Function evaluation involves substituting the given values into a function to determine a specific result. In the example provided:

• The function \(V(x,y) = \pi x^2 y\) calculates the volume of a cylinder using radius \(x\) and height \(y\).
• We substitute \(x = 2\) and \(y = 5\) into the function to find \(V(2,5)\).

This process is important in problem-solving, as it transforms abstract equations into tangible outputs. By using specific numbers, we can determine exact values, like calculating the cylinder's volume, in this case, to be approximately 62.83 cubic units. This demonstrates how functions can provide critical insights in both academic exercises and real-life applications such as engineering calculations.