Problem 373 - Precalculus

Question

Use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(4 x^{3}+12 x^{2}-15 x-50\right),\) radius is \(2 x+5.\)

Step-by-Step Solution

Answer

The height of the cylinder is \( h = \frac{4x^3 + 12x^2 - 15x - 50}{4x^2 + 20x + 25} \).

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

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

Step 2: Identify given values and expressions

We are given that the volume \( V \) is \( \pi(4x^3 + 12x^2 - 15x - 50) \) and the radius \( r \) is \( 2x + 5 \). We want to express the height \( h \) in terms of \( x \).

Step 3: Solve the volume formula for height (h)

Rearrange the formula for the volume of a cylinder to solve for \( h \): \( h = \frac{V}{\pi r^2} \).

Step 4: Calculate \( r^2 \)

Substitute the expression for \( r \) into \( r^2 \): \( r = 2x + 5 \), so \( r^2 = (2x + 5)^2 = 4x^2 + 20x + 25 \).

Step 5: Substitute values into the height formula

Substitute \( V \) and \( r^2 \) into the height formula: \( h = \frac{\pi(4x^3 + 12x^2 - 15x - 50)}{\pi (4x^2 + 20x + 25)} \).

Step 6: Simplify the expression for \( h \)

Cancel out \( \pi \) from the numerator and denominator, leaving \( h = \frac{4x^3 + 12x^2 - 15x - 50}{4x^2 + 20x + 25} \). This is the algebraic expression for the height of the cylinder.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in mathematics. It involves rearranging equations and expressions to simplify and solve for a particular variable.
In the context of calculating the height of a cylinder, we start with the volume formula \[ V = \pi r^2 h \]
where:

• \( V \) represents the volume,
• \( r \) is the radius,
• \( h \) is the height of the cylinder.

Our goal here is to solve for the height \( h \). To do this, we rearrange the formula:\[ h = \frac{V}{\pi r^2} \]This formula transformation is a perfect example of algebraic manipulation, as it isolates the variable \( h \) on one side for easier computation as we apply given values.

Cylinder Height Expression

Once we have manipulated our formula to solve for \( h \), the next step is expressing this height in terms of given values.
From the original problem, the volume \( V \) is provided as \( \pi(4x^3 + 12x^2 - 15x - 50) \) and the radius as \( 2x + 5 \).
To find the height, we calculate \( r^2 \):

• Given \( r = 2x + 5 \)
• \( r^2 = (2x + 5)^2 \)
• \( r^2 = 4x^2 + 20x + 25 \)

We then substitute \( V \) and \( r^2 \) into the height formula:\[ h = \frac{\pi(4x^3 + 12x^2 - 15x - 50)}{\pi (4x^2 + 20x + 25)} \]
By canceling the common factor \( \pi \), the expression simplifies to: \[ h = \frac{4x^3 + 12x^2 - 15x - 50}{4x^2 + 20x + 25} \] This expression represents the cylinder's height based on the polynomial expressions in the numerator and denominator.

Polynomial Expressions

Polynomial expressions are algebraic expressions that include variables raised to whole-number exponents combined using addition, subtraction, or multiplication.
In this problem, both the numerator and denominator of the height formula involve polynomial expressions.

• The numerator \( 4x^3 + 12x^2 - 15x - 50 \) is a third-degree polynomial, indicating it was originally the volume representation minus the \( \pi \).
• The denominator \( 4x^2 + 20x + 25 \) is a second-degree polynomial from the square of the radius.

Understanding these polynomial expressions is vital as they determine the complexity and behavior of the resulting \( h \) expression.
Such expressions could be evaluated for specific values of \( x \) to find the height for various scenarios, showcasing the flexibility polynomials provide in modeling mathematical properties.