Problem 374 - Precalculus

Question

Use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(3 x^{4}+24 x^{3}+46 x^{2}-16 x-32\right),\) radius is \(x+4.\)

Step-by-Step Solution

Answer

The height \( h \) is \( \frac{3x^4 + 24x^3 + 46x^2 - 16x - 32}{(x+4)^2} \).

Step 1: Recall the volume formula for a cylinder

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

Step 2: Set up the equation for height

Given the volume \( V = \pi(3x^4 + 24x^3 + 46x^2 - 16x - 32) \) and the radius \( r = x + 4 \), we can use the formula \( V = \pi r^2 h \). Substitute the known values into the formula to identify the expression for height: \[ \pi(3x^4 + 24x^3 + 46x^2 - 16x - 32) = \pi (x + 4)^2 h \].

Step 3: Cancel \(\pi\) from both sides

Divide both sides of the equation by \( \pi \) to simplify: \[ 3x^4 + 24x^3 + 46x^2 - 16x - 32 = (x + 4)^2 h \].

Step 4: Expand the squared radius

Calculate \((x + 4)^2\) separately to manage the expression. Expanding, we have: \[ (x + 4)^2 = x^2 + 8x + 16 \].

Step 5: Solve for height \( h \)

Substitute \( (x + 4)^2 \) into the equation, and then isolate \( h \) by dividing the polynomial on the left by the resulting quadratic expression: \[ h = \frac{3x^4 + 24x^3 + 46x^2 - 16x - 32}{x^2 + 8x + 16} \].

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill when working with equations and expressions. It allows you to rearrange, simplify, or solve for a variable, transforming complex expressions into more manageable forms. In the context of finding the height of a cylinder, we begin by using the standard volume formula for a cylinder, \( V = \pi r^2 h \). Given variables in the problem, such as the volume in polynomial form and a specific radius, we substitute these into the formula and manipulate the equation to isolate and solve for height \( h \).

Here’s a quick overview of how to apply algebraic manipulation in this context:

• Substitute the given volume polynomial and value of \( r \) into the cylinder volume formula.
• Cancel out common terms such as \( \pi \) from both sides to simplify the equation.
• Expand expressions where necessary to prepare for division or further simplification, which helps in isolating \( h \).

By breaking down and rearranging the terms, we create a pathway to solving for the desired variable, in this case, the height \( h \) of the cylinder.

Cylinder Height Calculation

Calculating the height of a cylinder when given its volume and radius involves careful algebraic manipulation. The primary method is to first set up the equation using the volume formula \( V = \pi r^2 h \), and then solve for \( h \).

Let's summarize the steps to calculate the height as per the exercise:

• Set the given volume expression equal to \( \pi r^2 h \).
• Cancel \( \pi \) from both sides if applicable to simplify the equation.
• Replace \( r^2 \) with the expanded form \((x+4)^2 = x^2 + 8x + 16\).
• Finally, divide the entire polynomial expression representing the volume by the squared term \((x+4)^2\) to isolate \( h \).

The key here is to systematically simplify the polynomial expression, leading to an accurate representation of height. This step-by-step breakdown helps you visualize the height as a function of the irradiated area of the cylinder's circular base.

Polynomial Division

Polynomial division is an essential technique in simplifying expressions and solving equations involving polynomials. In this exercise, polynomial division helps find the height \( h \) by dividing the given volume polynomial by the squared radius expression.

Here’s how polynomial division is applied:

• Start by identifying the dividend, which is the polynomial expression of the volume \( 3x^4 + 24x^3 + 46x^2 - 16x - 32 \).
• The divisor, in this case, is the squared radius \( x^2 + 8x + 16 \), derived from \((x+4)^2\).
• Perform polynomial long division or synthetic division to divide the volume polynomial by the squared radius.
• The quotient obtained represents the height \( h \), satisfying the equation for cylinder volume.

Understanding polynomial division allows one to handle varying coefficients and degrees of polynomials, facilitating the process of isolating the height of a cylinder from its volume equation. This process is not only crucial in algebra but also in practical real-world scenarios where relationships between quantities need to be expressed clearly.