Problem 16 - Algebra for College Students

Question

\text { Solve each of the following for the indicated variable. } $$ V=B h \quad \text { for } h $$

Step-by-Step Solution

Answer

The solution is \( h = \frac{V}{B} \).

Step 1: Understand the Equation

The equation given is \( V = B \cdot h \). This equation suggests that \( V \) (volume) is the product of \( B \) (base area) and \( h \) (height). Our goal is to solve for \( h \).

Step 2: Isolate the Variable

To solve for \( h \), we need to isolate \( h \) on one side of the equation. Start by dividing both sides of the equation by \( B \):\[ \frac{V}{B} = \frac{B \cdot h}{B} \]

Step 3: Simplify the Equation

Simplifying the right side, the \( B \) terms cancel each other out:\[ \frac{V}{B} = h \]

Step 4: Write the Solution

The final solution after simplifying is:\[ h = \frac{V}{B} \].

Key Concepts

Isolating a Variable

Isolating a variable is like trying to focus on one idea by itself. It involves getting the variable you need on one side of the equation, completely alone. Think of it as untangling a knot so that the variable stands out clearly. In our example, we started with the equation \( V = B \, h \) and wanted to "free" \( h \). To do this, you might need to do some division, subtraction, or other operations, depending on the equation.

Here’s a quick rundown of steps you often take when isolating a variable:

• **Identify:** Determine what variable you need to isolate. In our equation, it’s \( h \).
• **Operate:** Use addition, subtraction, multiplication, or division to move everything else to the opposite side.
• **Simplify:** Make sure the variable stands alone, just like we have \( h = \frac{V}{B} \).

It’s just like solving a puzzle by rearranging pieces until you see the picture you want.

Algebraic Manipulation

Algebraic manipulation sounds fancy, but it's simply doing things step-by-step to change an equation while keeping it balanced. Imagine you're a chef making a recipe, and you need to change some ingredients. You do it carefully, ensuring the recipe still tastes good!

In the example \( V = B \, h \), algebraic manipulation is used to move things around to solve for \( h \):

• **Balance Both Sides:** When you divide or multiply on one side, do it to the other side too!
• **Maintain Structure:** Just like keeping ingredients balanced in a recipe, ensure the equation’s form stays balanced.
• **Cancel Out:** Use operations to "cancel out" parts you don’t need on the variable's side, like how we divided by \( B \) and removed it.

It's all about knowing what moves make the equation stay true and finding your variable.

Formula Rearrangement

Formula rearrangement is like redecorating a room. You still have the same items, but you've placed them differently to make things flow better. The purpose is to highlight or focus on something specific, like a particular variable. In math, this means reorganizing the equation to bring the variable of interest to the forefront.

Using our equation \( V = B \, h \) as an example, we wanted \( h \) highlighted and central. Here’s how formula rearrangement works in practice:

• **Target the Variable:** Decide which variable you want to focus on. Here it’s \( h \).
• **Reorganize:** Slide other parts of the equation around to isolate your target. We moved \( B \) by dividing to one side to simplify.
• **Clear Presentation:** Make sure the final equation is easy to understand and shows what the formula means, just like showing \( h = \frac{V}{B} \).

Essentially, rearranging formulas helps you see relationships clearly, letting you spot how variables interlink.