Problem 17 - Algebra for College Students

Question

Solve each of the following for the indicated variable. \(V=B h \quad\) for \(h \quad\) (Volume of a prism)

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

We are given the equation for the volume of a prism: \( V = B h \), where \( V \) is the volume, \( B \) is the base area, and \( h \) is the height. We need to solve this equation for the variable \( h \).

Step 2: Isolate the Variable \( h \)

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

Step 3: Simplify the Equation

Simplification of \( \frac{B h}{B} \) results in \( h \). So, after simplification, we have:\[ h = \frac{V}{B} \]

Step 4: Write the Solution

The solution for the variable \( h \) in terms of \( V \) and \( B \) is:\[ h = \frac{V}{B} \]

Key Concepts

Solving Equations

Solving equations is an essential skill in algebra. It involves finding the value of a variable that makes an equation true. The process starts with having an algebraic expression and manipulating it until the variable is isolated. For example, consider an equation like \( V = B h \), where \( V \) is the result we want to express in terms of another variable. Here, the task is to express the variable \( h \) (height) in terms of \( V \) (volume) and \( B \) (base area). The steps lead to isolating \( h \) by performing operations that simplify this goal. Remember, each operation done on one side of an equation should also be done on the other side to maintain equality. When the variable gets isolated, the equation is solved.

Variable Isolation

Variable isolation is a method used to find the value of an unknown variable in equations. It involves rearranging the equation using mathematical operations so the variable you are solving for stands alone on one side of the equation. For instance, when given the equation \( V = B h \), to determine \( h \), we need to isolate it by dividing both sides by \( B \).

• Start with \( V = B h \).
• Divide both sides by \( B \) to get \( \frac{V}{B} = h \).

These steps look simple, but they are crucial to unravel how a variable depends on others in any equation. Variable isolation skills are foundational because they form the basis of solving many types of algebraic equations.

Volume of a Prism

The volume of a prism can be thought of as the space contained within it. It is calculated based on the base area and the height of the prism. The formula is given by \( V = B h \), where \( V \) denotes the volume, \( B \) is the base area, and \( h \) is the height. This simple formula applies to any prism, whether it's a rectangular prism, a triangular prism, or any other shape with a consistent cross-section.

• The base area \( B \) depends on the dimensions of the shape that forms the base.
• The height \( h \) is the perpendicular distance between the base and the top of the prism.

When solving for \( h \), you can rearrange the formula by dividing the volume by the base area, revealing how volume is distributed in the prism to form its height. Understanding this relationship is crucial when working with three-dimensional objects.