Problem 73 - College Algebra

Question

For the following exercises, 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 is expressed as \( h = \frac{3x^4 + 24x^3 + 46x^2 - 16x - 32}{x^2 + 8x + 16} \).

Step 1: Understand the Formula for Volume of a Cylinder

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

Step 2: Substitute Given Values into the Formula

We are given the volume \( V = \pi(3x^4 + 24x^3 + 46x^2 - 16x - 32) \) and the radius \( r = x + 4 \). Substitute these into the formula: \[ \pi(3x^4 + 24x^3 + 46x^2 - 16x - 32) = \pi (x+4)^2 h. \]

Step 3: Cancel Out \( \pi \) from Both Sides

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

Step 4: Expand the Square of the Radius

Expand \( (x+4)^2 \) to simplify the expression:\[ (x+4)^2 = x^2 + 8x + 16. \]

Step 5: Express the Height Algebraically

Substitute the expanded form back into the equation:\[ 3x^4 + 24x^3 + 46x^2 - 16x - 32 = (x^2 + 8x + 16)h. \] Solve for \( h \):\[ h = \frac{3x^4 + 24x^3 + 46x^2 - 16x - 32}{x^2 + 8x + 16}. \]

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can involve numbers, variables like \( x \), and arithmetic operations. It’s crucial in solving equations and understanding relationships between different quantities. For example, in our cylinder problem, the original expression for the volume is given as \( \pi(3x^4 + 24x^3 + 46x^2 - 16x - 32) \). This complex expression represents the volume of the cylinder involving a polynomial with \( x \) raised to various powers. By forming algebraic expressions, we can analyze the relationships and derive further useful expressions, such as the height of the cylinder in terms of the given volume and radius.

Radius and Height Relationship

In the context of a cylinder, the radius and height are key geometric parameters that determine the volume. The formula \( V = \pi r^2 h \) highlights the interaction between these quantities. The volume \( V \) is directly proportional to both the square of the radius \( r^2 \) and the height \( h \). For a given volume, this relationship allows us to express one dimension in terms of the others. For instance, knowing the radius \( r = x + 4 \) and the algebraic volume expression, we can substitute and manipulate the equation to isolate and express the height \( h \) algebraically. This highlights the interdependence of the dimensions and the ability to derive one dimension from others using algebra.

Expanding Expressions

Expanding expressions involves multiplying and simplifying to transform a product of simpler expressions into a polynomial. In our exercise, we expand \( (x+4)^2 \) as part of solving for the height. This step results in \( x^2 + 8x + 16 \).

• Distribution: Each term in one bracket multiplies each term in the other.
• Combining Like Terms: Add terms with the same variable powers.

This process simplifies the overall equation and is essential in isolating \( h \). Without expanding expressions, such algebraic manipulations would be cumbersome and difficult to solve. Proper expansion ensures that the expressions are straightforwardly manipulatable leading to easier problem-solving and algebraic reasoning.