Problem 20 - Algebra for College Students

Question

$$ V=\frac{1}{3} B h \quad \text { for } B $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

We have the formula for the volume of a cone or pyramid, which is given by \( V = \frac{1}{3} B h \), where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height. The task is to solve this equation for \( B \).

Step 2: Isolate B on One Side

To isolate \( B \), we need to get rid of the fraction and the height \( h \). Start by multiplying both sides of the equation by 3 to eliminate the fraction: \[ 3V = B h \].

Step 3: Solve for B

Now, divide both sides by \( h \) to solve for \( B \): \[ B = \frac{3V}{h} \].

Key Concepts

Solving for a variable

Solving for a variable involves rearranging an equation so that the variable of interest, often abbreviated like \( x \), \( y \), or in our case \( B \), stands alone on one side of the equation. This process helps us find the value of the unknown variable when the other values are known.

To do this, you need to understand the steps involved:

• Identify which variable you need to solve for.
• Use algebraic operations to move other terms away from this variable.
• Simplify the equation if necessary to make it clearer.

This process of solving for a variable is fundamental in algebra and is used in various applications, including calculus, statistics, and everyday word problems. It is a vital skill for breaking down complex problems into simple, solvable parts.

Volume formula

The volume formula in question refers specifically to three-dimensional shapes like cones or pyramids. The formula given is \( V = \frac{1}{3} B h \), where:

• \( V \) stands for the volume of the shape.
• \( B \) is the area of the base of the shape.
• \( h \) is the height of the shape.

This formula is derived from the concept of a prism, where the volume is equal to the base area multiplied by the height. However, since cones and pyramids taper towards the top, their volume is only a third of what it would be if they were acting as a full prism. This division by 3 represents how these shapes "shrink" as they go up.

Isolating variables

The process of isolating a variable is a key step in solving equations. It involves performing operations to both sides of the equation to ensure the variable you're solving for is by itself on one side of the equals sign. In our case, isolating \( B \) involved the following steps:

• Eliminate any fractions by multiplying both sides by the denominator. Here, we multiplied both sides by 3 to rid the equation of \( \frac{1}{3} \).
• Use division to separate the variable from any coefficients. By dividing both sides by \( h \), we further isolated \( B \) making it \( B = \frac{3V}{h} \).

Isolating the variable correctly is crucial because any mistakes in this process can lead to incorrect solutions, which is why careful attention to each algebraic step matters.

Mathematical equations

Mathematical equations are statements that express the equality between two expressions. They are the backbone of algebra and represent real-world phenomena in a mathematical form, allowing for the calculation and prediction of outcomes.

Equations consist of variables, constants, and arithmetic operations. The goal is often to solve these equations to find the value of unknown variables. This act involves logical steps which may include:

• Balancing: Ensuring both sides of the equation remain equal as you manipulate them.
• Simplifying: Reducing the equation to its simplest form.
• Checking: Verifying the solution by substituting the found values back into the original equation.

Understanding and working with equations is a fundamental aspect of algebra, which extends to more advanced mathematics and various scientific fields.