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Chapter 2: Problem 85

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

Short Answer

Expert verified
The solution for \(h\) is \( h = \frac{3V}{\pi r^{2}} \).

Step by step solution

01

Understand the Formula

The formula given is the volume of a cone, where \(V\) represents the volume, \(r\) represents the radius of the base, and \(h\) represents the height. The formula is: \[ V = \frac{1}{3} \times \pi r^{2} \times h \]
02

Isolate the Variable

To solve for \(h\), isolate the variable on one side of the equation. Start by multiplying both sides by 3 to get rid of the fraction: \[ 3V = \pi r^{2} h \]
03

Solve for h

Divide both sides by \( \pi r^{2} \) to solve for \(h\): \[ h = \frac{3V}{\pi r^{2}} \]

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Key Concepts

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

algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, we use letters (often called variables) to represent unknown values or values that can change. For example, in the equation \(V = \frac{1}{3} \pi r^{2} h\), the variables are \(V\), \(r\), and \(h\). Understanding how to manipulate these variables to solve for a specific one is key in algebra. This exercise involves isolating the variable \(h\). Using algebraic principles, we can rearrange the formula so that \(h\) is by itself on one side of the equation.
volume of a cone
The volume of a cone can be calculated using a specific formula: \(V = \frac{1}{3} \pi r^{2} h\). In this formula, \(V\) represents the volume of the cone, \(r\) is the radius of its base, and \(h\) is the height of the cone. Since a cone is a three-dimensional shape, its volume is measured in cubic units. The formula essentially indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height. Understanding how to apply this formula helps in solving for any of these variables when the other two are known.
isolating variables
Isolating variables involves manipulating the equation so that a specific variable stands alone on one side of the equation. This is essential for solving equations. In the given exercise, we need to solve for \(h\). To do this, we perform operations to isolate \(h\) on one side of the equation. The initial formula is \(V = \frac{1}{3} \pi r^{2} h\). First, we eliminate the fraction by multiplying both sides of the equation by 3, resulting in \(3V = \pi r^{2} h\). Next, we divide both sides by \(\pi r^{2}\), leading to \(h = \frac{3V}{\pi r^{2}}\). This demonstrates the process of isolating \(h\).
step-by-step solution
Breaking down complex problems into step-by-step solutions helps in understanding and solving them independently. To solve the given formula for \(h\), we follow these steps:
  • Step 1: Understand the Formula - Identify the variables and the formula \(V = \frac{1}{3} \pi r^{2} h\).

  • Step 2: Isolate the Variable - Multiply both sides by 3 to remove the fraction: \(3V = \pi r^{2} h\).

  • Step 3: Solve for \(h\) - Divide both sides by \(\pi r^{2}\) to get \(h = \frac{3V}{\pi r^{2}}\).

By understanding each step and its purpose, we can effectively solve for the needed variable.

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Most popular questions from this chapter

Two angles whose measures sum to \(90^{\circ}\) are \(\longrightarrow\) angles. Two angles whose measures sum to \(180^{\circ}\) are \(\longrightarrow\) angles.

A one-cup serving of pineapple juice has 9 more than three times as many calories as a one-cup serving of tomato juice. Servings of the two juices contain a total of 173 calories. How many calories are in a serving of each type of juice? (Data from U.S. Agriculture Department.)

If five times the lesser of two consecutive integers is added to three times the greater, the result is \(59 .\) Find the integers.

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The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In triangle \(A B C,\) the measure of angle \(A\) is \(141^{\circ}\) greater than the measure of angle \(B\). The measure of angle \(B\) is the same as the measure of angle \(C\). Find the measure of each angle.
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