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Chapter 2: Problem 79

Solve each formula for the specified variable. $$ V=\frac{1}{3} \pi r^{2} h \text { for } h $$

Short Answer

Expert verified
The short answer is: \( h = \frac{3V}{\pi r^2} \)

Step by step solution

01

Understand the formula

The given formula is the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \] We need to solve this formula for the height (\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of a Cone
To understand how to solve the given formula for height, let's first understand the volume of a cone. The volume (V) of a cone can be found using the formula \[ V = \frac{1}{3} \pi \ r^{2} h \] In this formula:
  • \(V\) represents the volume of the cone
  • \(r\) is the radius of the base
  • \(h\) is the height of the cone
Explanation: The formula calculates the volume by multiplying the area of the base of the cone, which is a circle (\( \pi \ r^{2} \)), by the height (\( h \)) and then dividing by 3. This formula is derived from the volume of a cylinder, adjusted by the fact that a cone is essentially one-third of a cylinder with the same base and height.
Algebraic Manipulation
Next, we use algebraic manipulation to solve the given formula for the height (\( h \)).
This means rearranging the formula so that \( h \) is isolated on one side of the equation.
Given formula: \[ V = \frac{1}{3} \pi \ r^{2} h \]
Step 1: Multiply both sides of the equation by 3 to eliminate the fraction:
\[ 3V = \ pi \ r^{2} h \]
Step 2: Divide both sides of the equation by \pi \ r^{2} to isolate \( h \):
\[ h = \frac{3V}{ \pi \ r^{2}} \]
Explanation: These steps use basic algebra to rearrange the formula.
  • First, we eliminate the fraction by multiplying by 3.
  • Second, we solve for \( h \) by dividing by all other variables and constants attached to \( h \).
This type of manipulation is powerful and can be used to isolate and solve for any variable in an equation.
Geometry Formulas
Understanding geometry formulas is crucial in solving related problems accurately.
Common Geometry Formulas:
  • Volume of a cylinder: \[ V = \pi \ r^{2} h \]
  • Volume of a sphere: \[ V = \frac{4}{3} \pi \ r^{3} \]
  • Area of a circle: \[ A = \pi \ r^{2} \]
Explanation: Geometry heavily relies on formulas to calculate different properties like areas, volumes, and angles.
Knowing the standard formulas allows you to solve more complex problems by breaking them down into simpler parts. Practicing these formulas makes it easier to remember and apply them correctly in exams and homework.

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