Problem 41 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ V=\frac{1}{3} B h \text { for } B $$

Step-by-Step Solution

Answer

The solution for \( B \) is \( B = \frac{3V}{h} \).

Step 1: Understand the Given Formula

The given formula is the volume of a pyramid, which is expressed as \( V = \frac{1}{3} B h \), where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height. We need to solve for \( B \).

Step 2: Eliminate the Fraction

To isolate \( B \), first eliminate the fraction by multiplying both sides of the equation by 3. This gives \( 3V = B h \).

Step 3: Solve for \( B \)

Now that we have \( 3V = B h \), divide both sides of the equation by \( h \) to solve for \( B \). This results in \( B = \frac{3V}{h} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a critical skill when solving equations or rearranging formulas. It involves using operations to transform equations into a simpler or more useful form. In the given exercise, we started with the formula for the volume of a pyramid, expressed as \( V = \frac{1}{3} B h \). The task was to solve for the base area \( B \).

Here are some steps involved in algebraic manipulation that were used in this exercise:

• **Understanding operations:** Recognizing what operations are acting on the variable you need to solve for. In this case, multiplication by a fraction.
• **Applying inverse operations:** We used multiplication to eliminate the fraction by multiplying both sides by 3.
• **Maintaining equation balance:** Any operation done on one side of the equation must be done on the other side to keep it balanced.

Mastering these techniques allows for a straightforward path to isolating and solving for the target variable.

Formula Rearrangement

Rearranging formulas is essential when you require a particular variable in isolation. It often involves a series of algebraic steps to maintain the integrity of the equation while transforming it.

In the context of our problem:

• **Identifying the target variable:** Initially, the problem requires solving for \( B \), which is the base area of the pyramid.
• **Simplifying:** We simplified the fraction by multiplying through by 3, converting \( V = \frac{1}{3} B h \) into \( 3V = B h \).
• **Isolating the variable:** The final step involved dividing both sides by \( h \), resulting in \( B = \frac{3V}{h} \). This completely isolates \( B \).

Through formula rearrangement, solving for any variable within a multi-variable equation becomes a structured and predictable process. It helps in ensuring that we get to an expression where the variable of interest is neatly isolated.

Geometry Applications

Geometry often involves equations that describe shapes and volumes, much like in our exercise involving the pyramid. Understanding the geometric context of a problem can guide the algebraic solution as it defines what each symbol in the equation represents.

In the pyramid volume formula \( V = \frac{1}{3} B h \):

• **Volume Definition:** The volume \( V \) gives a sense of the 3D space occupied by the pyramid.
• **Base Area \( B \):** This represents the area of the base of the pyramid, which is multiplied by the height to scale up to the volume.
• **Height \( h \):** The vertical distance from the base to the apex (tip) of the pyramid.

In this geometric application, recognizing that reshuffling the formula \( B = \frac{3V}{h} \) allows for directly calculating the base area when the volume and height are known. It can simplify practical tasks, like determining the base size needed for a pyramid given certain volume or height constraints.