Problem 72 - 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 \(\pi\left(4 x^{3}+12 x^{2}-15 x-50\right),\) 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: Understanding the Volume Formula

The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cylinder. We are given the volume \( V = \pi (4x^3 + 12x^2 - 15x - 50) \) and radius \( r = 2x + 5 \).

Step 2: Applying the Cylinder Volume Formula

Substitute the given radius \( r \) into the volume formula. The equation becomes: \[ \pi (4x^3 + 12x^2 - 15x - 50) = \pi (2x + 5)^2 h \].

Step 3: Simplifying the Radius Squared

Calculate \((2x + 5)^2\) by expanding it: \((2x + 5)^2 = (2x + 5)(2x + 5) = 4x^2 + 20x + 25\). Substitute this back into the equation to get: \[ \pi (4x^3 + 12x^2 - 15x - 50) = \pi (4x^2 + 20x + 25) h \].

Step 4: Canceling \(\pi\) from Both Sides

Since \(\pi\) is a common factor on both sides of the equation, cancel \(\pi\) out, resulting in: \[ 4x^3 + 12x^2 - 15x - 50 = (4x^2 + 20x + 25) h \].

Step 5: Solving for Height \(h\)

Isolate \(h\) by dividing both sides of the equation by \(4x^2 + 20x + 25\): \[ h = \frac{4x^3 + 12x^2 - 15x - 50}{4x^2 + 20x + 25} \]. This expression represents the height of the cylinder in terms of \(x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental in mathematics. They consist of variables, constants, and algebraic operations like addition and multiplication. In our exercise, the expression for the volume of the cylinder is an algebraic expression. We have

• Variables: These are represented by the letter \(x\) in the expression.
• Constants: The numbers like \(4\), \(12\), \(15\), and \(50\) are constants that do not change.
• Operations: Involves addition, subtraction, and multiplication by variables or constants.

These expressions allow us to depict real-world problems mathematically. By manipulating them correctly, we can uncover unknowns such as the height of a cylinder. Remember, simplifying and rearranging algebraic expressions accurately is crucial to solving complex mathematical problems.

Polynomial Division

Polynomial division, similar to long division in arithmetic, helps reduce complex polynomial expressions into simpler forms. In this exercise, we needed to divide a cubic polynomial by a quadratic polynomial.
The cubic polynomial here is \(4x^3 + 12x^2 - 15x - 50\). The quadratic polynomial is \(4x^2 + 20x + 25\). Dividing these polynomials allows us to solve for the height \(h\) in a more simplified manner.

• Start by comparing leading terms: Divide the first term of the numerator by the first term of the denominator.
• Multiply the entire divisor by this quotient and subtract from the original polynomial.
• Repeat the process with the remainder until no further division is possible.

By understanding polynomial division, you can work through such expressions systematically to simplify or solve equations.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. In our exercise, this was done to find the height \(h\) of the cylinder.
The key steps include:

• Setting up the equation: Using information given like the volume and radius of the cylinder.
• Substituting known values: The radius and the expanded form \((4x^2+20x+25)\) were used.
• Canceling common terms: Here, \(\pi\) was canceled from both sides to simplify the expression.
• Isolating the variable: Solving for \(h\) by dividing both sides of the equation by the polynomial \(4x^2 + 20x + 25\).

By applying these steps, equations can be solved effectively. This involves balancing the equation and correctly performing algebraic operations to isolate and solve for the desired variable.