Problem 20 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$h=\frac{S-2 \pi r^{2}}{2 \pi r} \text { for } S$$

Step-by-Step Solution

Answer

S = 2 \pi rh + 2 \pi r^2

Step 1: Isolate the term containing S

Start with the formula given: \[ h = \frac{S - 2 \pi r^2}{2 \pi r} \] Multiply both sides by \(2 \pi r \) to isolate the term containing S: \[ h \times 2 \pi r = S - 2 \pi r^2 \] So, \[ 2 \pi rh = S - 2 \pi r^2 \]

Step 2: Solve for S

Add \(2 \pi r^2\) to both sides of the equation to isolate S: \[ 2 \pi rh + 2 \pi r^2 = S \] Thus, \[ S = 2 \pi rh + 2 \pi r^2 \]

Key Concepts

algebraic manipulation

Algebraic manipulation is an essential skill in solving equations and formulas. It involves rearranging and simplifying expressions by performing operations such as addition, subtraction, multiplication, and division. In our exercise, we start with the formula for height, \( h = \frac{S - 2 \pi r^2}{2 \pi r} \), and employ algebraic manipulation to isolate and solve for the variable \( S \). This process makes the formula more versatile and usable in different contexts.
It's all about understanding the properties of equality and ensuring that the operations performed on one side of the equation are also carried out on the other side. You don't change the equation's meaning; you simply rearrange it to its different form.
These steps often involve:

• Identifying the terms and operations within the given equation
• Applying reversals of operations, such as multiplication and division, to move variables and constants around
• Ensuring balance by performing the same operation on both sides of the equation

This systematic approach helps in tackling more complex problems effectively.

isolating variables

Isolating variables is about getting the variable you are solving for by itself on one side of the equation. Think of it as 'untangling' the variable from the rest of the equation. In our example, we are solving for the variable \( S \).
We start with the equation \( h = \frac{S - 2 \pi r^2}{2 \pi r} \). To isolate \( S \), our first step is to eliminate the fraction by multiplying both sides by \( 2 \pi r \), thus simplifying our equation to \( h \times 2 \pi r = S - 2 \pi r^2 \).
Next, we add \( 2 \pi r^2 \) to both sides to completely isolate \( S \). This leaves us with \( S = 2 \pi rh + 2 \pi r^2 \).
These steps ensure that \( S \) is by itself and clearly defined in terms of the other variables in the equation.

formula rearrangement

Formula rearrangement involves changing the structure of a formula to solve for a specific variable. This is often necessary in science, engineering, and mathematics, where formulas represent relationships between different variables. In our provided exercise, we rearranged the formula \( h = \frac{S - 2 \pi r^2}{2 \pi r} \) to solve for \( S \).
The process involves:

• Carefully applying algebraic operations to shift terms around
• Isolating the desired variable on one side of the equation
• Ensuring every step maintains the original equation's balance and equality

It's an important skill because it allows one formula to be adapted for different purposes. Instead of memorizing multiple variations of formulas, students can learn to rearrange a given formula to solve for any of its variables. This flexibility is key to mastering algebra and higher-level mathematics.