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Chapter 10: Problem 43

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

Short Answer

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

Step by step solution

01

- Understand the Formula

Before doing anything, it's crucial to understand what is being asked. This formula relates four quantities: \( r \) (radius), \( h \) (height), \( V \) (volume), and \( \pi \) (a constant equal to approximately 3.14). The task is to solve for \( V \), which means that we need to isolate \( V \) on one side of the equation.
02

- Square both sides

Start by squaring both sides of the equation to get rid of the square root. That leads to the equation: \( r^2 = \frac{3V}{\pi h} \)
03

- Solve for V

Isolate \( V \) on one side of the equation by multiplying both sides by \( \pi h \). This gives the formula for the volume \( V \) of a cylinder: \( V = \frac{r^2 \pi h}{3} \)

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Key Concepts

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

Solving Equations
Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. Consider the formula provided: \( r = \sqrt{\frac{3V}{\pi h}} \). Here, our goal is to solve for \( V \), the volume. This involves a series of manipulations to rearrange the equation so that \( V \) is isolated on one side. Most importantly, when solving equations:
  • Identify the variable you need to solve for.
  • Perform the same operation on both sides to maintain equality.
  • Aim to gradually isolate the variable through strategic steps.
Solving the equation step-by-step gives clarity and prevents errors. In this case, squaring both sides helps remove the square root, simplifying the equation and bringing us closer to isolating \( V \).
Formula Rearrangement
Formula rearrangement involves changing the structure of an equation to solve for a particular variable. In our exercise, after squaring both sides, our equation transforms from \( r = \sqrt{\frac{3V}{\pi h}} \) to \( r^2 = \frac{3V}{\pi h} \). The transformation is a crucial step that helps us rearrange the formula to solve for \( V \).
  • Rearrangement often involves performing inverse operations like addition/subtraction or multiplication/division.
  • Keep the balance by executing operations on both sides evenly.
  • Be meticulous when handling constants and coefficients.
This rearrangement process is vital as it steadily morphs the equation to highlight the variable we are interested in isolating.
Isolation of Variables
Isolation of variables is where we get the variable of interest by itself on one side of the equation. Once we arrived at \( r^2 = \frac{3V}{\pi h} \), the next step was isolating \( V \). By multiplying both sides by \( \pi h \), we succeeded in freeing \( V \), leading us to the formula \( V = \frac{r^2 \pi h}{3} \).
  • Isolation often involves using inverse operations: e.g., division can isolate a multiplied variable.
  • This process makes it easier to find the value of the isolated variable if all other values are known.
  • Double-check each step to confirm that operations have been applied correctly.
Successfully isolating the variable makes the formula work-ready, allowing us to substitute in values and solve for the needed variable effortlessly.

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Most popular questions from this chapter

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{3}{\sqrt[4]{x^{5} y^{3}}}$$

In Exercises \(129-132\), determine if each operation is performed correctly by graphing the function on each side of the equation with your graphing utility. Use the given viewing rectangle. The graphs should be the same. If they are not, correct the right side of the equation and then use your graphing utility to verify the correction. $$\begin{aligned} &\frac{3}{\sqrt{x+3}-\sqrt{x}}=\sqrt{x+3}+\sqrt{x}\\\ &[0,8,1] \text { by }[0,6,1] \end{aligned}$$

Explain how to perform this multiplication: \(\sqrt{2}(\sqrt{7}+\sqrt{10})\)

In Exercises \(137-140\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some irrational numbers are not complex numbers.

Exercises \(147-149\) will help you prepare for the material covered in the next section. Solve: \(26-11 x=16-8 x+x^{2}\)
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