Problem 86 - College Algebra with Modeling and Visualization

Question

Solve the equation for the specified variable. $$ V=2 \pi r h+\pi r^{2} \text { for } h $$

Step-by-Step Solution

Answer

The solution for \( h \) is \( h = \frac{V - \pi r^2}{2 \pi r} \).

Step 1: Subtract the Constant Term

Start by isolating the term with the variable of interest. Subtract \( \pi r^2 \) from both sides of the equation to remove the constant term: \[ V - \pi r^2 = 2 \pi r h \].

Step 2: Divide by the Coefficient

Next, solve for \( h \) by dividing both sides of the equation by \( 2 \pi r \): \[ h = \frac{V - \pi r^2}{2 \pi r} \].

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra and mathematics. The goal is to find the unknown value that makes an equation true. Equations typically have one or more variables and constants. In our example, we are focused on isolating a specific variable.
To solve equations effectively:

• First, understand the equation you are dealing with and identify the variable you need to solve for, in this case, 'h'.
• Next, perform operations that simplify the equation while keeping it balanced. This involves applying the same operation to both sides.
• Use inverse operations to undo operations and isolate the variable.

Effective equation-solving requires patience and careful manipulation of the equation to achieve a simplified form.

Variable Isolation

Variable isolation is the process of rearranging an equation to get one variable alone on one side of the equation. This is essential in solving algebraic equations since it allows you to determine the variable's value directly.
To isolate the variable:

• Identify the specific term(s) containing the variable of interest.
• Remove any constants or terms not involving the variable by performing operations such as addition or subtraction.
• Apply inverse operations to eliminate coefficients attached to the variable. This may require multiplication or division.

In our example, we subtract \( \pi r^2 \) first to isolate the term with 'h', demonstrating the principle of getting the variable by itself.

Division in Equations

Division in equations is used to simplify expressions and isolate variables. It involves dividing both sides of an equation by the same non-zero number to maintain the equality.
In the equation \( V - \pi r^2 = 2 \pi r h \), we need to isolate 'h'.

• Recognize that 'h' is multiplied by \( 2 \pi r \). To undo this multiplication, divide by \( 2 \pi r \) on both sides.
• This operation yields the equivalent equation \( h = \frac{V - \pi r^2}{2 \pi r} \).
• It is crucial to remember dividing by zero is not possible, so ensure \( 2 \pi r e 0 \).

Using division efficiently helps us understand relationships between variables and constants and makes solving equations straightforward.