91Ó°ÊÓ

An algebraic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations.
In our cylinder problem, we are tasked with expressing the height of a cylinder using an algebraic expression.
To find this expression, we need to manipulate formula components algebraically.
Specifically, the formula for the volume of a cylinder is key in deriving the expression.
In our problem, the volume was given as a complex polynomial \( \pi(25x^3 - 65x^2 - 29x - 3) \).
Solving for the height means rearranging the formula to express the height as:

• Identify and cancel like terms.
• Rearrange the equation to isolate the desired variable, in this case, the height.

These operations yield the algebraic expression for the height:
\[ h = \frac{25x^3 - 65x^2 - 29x - 3}{25x^2 + 10x + 1} \] Each term in this expression serves a specific purpose, allowing the height to flexibly adjust based on the variable \(x\).

The height of a cylinder is a major component in determining its volume.
The volume formula is defined as \( V = \pi r^2 h \).
This means that manipulating either the radius or height impacts the volume. To determine the height algebraically, we must first express the volume equation in terms of height \( h \):

• Insert given values into the volume formula.
• Isolate the variable \( h \) to determine the height expression.

In our example, after substituting the values, we followed through cancelling and rearranging the equation:
From \( \pi (25x^3 - 65x^2 - 29x - 3) = \pi (5x + 1)^2 h \) to reach:
\[ h = \frac{25x^3 - 65x^2 - 29x - 3}{25x^2 + 10x + 1} \] This form shows how the height directly correlates to the cylinder's variable aspects, specifically \(x\).

The radius of a cylinder is another fundamental element related to its dimensions and volume.
For our problem, the radius has been defined as the algebraic term \( r = 5x + 1 \).
The role of the radius in the volume calculation is significant.
It is squared in the formula \( V = \pi r^2 h \), thus heavily influencing the cylinder's overall volume.
We use this radius to help express the height algebraically.
After substituting the radius into the volume formula, we needed to simplify:

• Calculate the square of the radius, resulting in \((5x + 1)^2\).
• Use this squared value to help solve for the height.

Since the radius is squared, any change in \( x \) directly affects both the height and the resulting volume.
This manipulation of the radius within the expression showcases its importance in algebraic problem-solving.