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Chapter 2: Problem 18

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

Short Answer

Expert verified
\( h = \frac{3V}{\pi r^{2}} \)

Step by step solution

01

- Understand the Formula

Identify the given formula and the variable to solve for. The formula is: \(V=\frac{1}{3} \pi r^{2} h\) We need to solve for \(h\).
02

- Isolate the Variable

To isolate \(h\), multiply both sides of the equation by 3 to get rid of the fraction:\[ 3V = \pi r^{2} h \]
03

- Divide by the Remaining Coefficients

Divide both sides of the equation by \(\pi r^{2}\) to isolate \(h\):\[ h = \frac{3V}{\pi r^{2}} \]

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Key Concepts

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

isolate the variable
In algebra, 'isolating the variable' means getting the variable you are solving for alone on one side of the equation. This process helps us find the value or expression for that variable. Let's study this concept using the given formula for the volume of a cone: \(V=\frac{1}{3} \pi \ r^{2} \ h\). We need to solve for \(h\).\When solving for \(h\), we start by identifying that \(h\) is currently multiplied by other constants and variables. To isolate \(h\), we will perform algebraic operations that 'undo' these multiplications/divisions.\Here's the process step-by-step: \
    \
  • First, multiply both sides by 3 to remove the fraction.
    • \
  • This gives us: \ \[ 3V = \pi \ r^{2} h \]
    • \
  • Next, divide both sides by \(\pi \ r^{2}\) to solve for \(h\).
    • \
\Now, \(h\) is isolated:\[ \h = \frac{3V}{\pi \ r^{2}}\]
volume of a cone
The volume of a cone is a measure of the amount of space inside the cone. The formula for the volume of a cone is: \ \(V=\frac{1}{3} \pi \ r^{2} \ h\).\Here's what each part of the formula represents: \ \
    \
  • \(V\) is the volume of the cone.
    • \
  • \(\pi\) (pi) is a constant (approximately 3.14159).
    • \
  • \(r\) is the radius of the cone's base.
    • \
  • \(h\) is the height of the cone.
    • \
\The formula calculates how much three-dimensional space the cone occupies. Understanding this formula is crucial when solving for any of the variables involved.
algebraic manipulation
Algebraic manipulation involves rearranging equations to solve for a specific variable. These steps often include adding, subtracting, multiplying, and dividing both sides of an equation.\Using the volume of a cone formula: \ \(V=\frac{1}{3} \pi \ r^{2} \ h\), we can manipulate it to solve for different variables.\In our case, we used algebraic manipulation to solve for \(h\): \ \
    \
  • First, we multiplied both sides by 3: \ \[3V = \pi \ r^{2} h\]
    • \
  • Then, divided by \(\pi \ r^{2}\) to get \(h\) alone: \[ \h = \frac{3V}{\pi \ r^{2}}\]
    • \
\Recognizing when and how to perform these operations is key to mastering algebra.

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Most popular questions from this chapter

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$2 y-7=3 y+1$$

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Solve each equation. $$2.365 x+3.694=14.8095$$
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