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Chapter 1: Problem 101

Solve for \(V: r=\sqrt{\frac{3 V}{\pi h}}\)

Short Answer

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

Step by step solution

01

Square both sides

Squaring both sides of the equation gets rid of the square root. \((r)^2 = \frac{3 V}{\pi h}\). The squared equation turns into \(r^2 = \frac{3V}{\pi h}\).
02

Rearrange the equation

Now, rearrange the equation to solve for \(V\). Multiply both sides by \(\pi h\) to isolate \(V\). The equation becomes \(V = \frac{\pi h r^2}{3}\).
03

Simplification

The equation is already simplified, so no other operation needs to be carried out.

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Key Concepts

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

Solving equations
In algebra, solving equations is all about finding the value of the unknown variable. For example, when dealing with a formula like \[ r = \sqrt{\frac{3V}{\pi h}} \]our goal is to isolate \( V \). This process involves manipulating both sides of the equation until \( V \) stands alone on one side. It鈥檚 like peeling back layers of an onion until you reach the core, or in this case, the solution.
  • Start by understanding what operations are needed to isolate the variable, such as addition, subtraction, multiplication, or division.
  • Always perform the same operation on both sides of the equation to maintain balance.
  • Keep track of each step as you go, ensuring nothing is overlooked.
Each step we take brings us closer to solving for \( V \) by peeling away complexity and arriving at a clearer and simpler equation. The key is patience and practice!
Variable manipulation
Variable manipulation is a crucial skill in simplifying and solving equations. Here, we want to manipulate the variables so that we can solve for \( V \). After squaring both sides of the equation, we have:\[ r^2 = \frac{3V}{\pi h} \]
To manipulate the variables:
  • Multiply both sides by \( \pi h \) to clear the fraction and bring \( V \) to one side.
  • This results in \( \pi h r^2 = 3V \), making it easier to solve for our variable \( V \).
  • Finally, divide by 3 to completely isolate \( V \).
Through these steps, variable manipulation helps to simplify the equation and make the solution more direct. It鈥檚 about using inverse operations and logical steps to reshape the equation into a solvable form.
Square root operations
Understanding square root operations is essential when dealing with equations involving roots. In the original problem, we see a square root:\[ r = \sqrt{\frac{3V}{\pi h}} \]
To remove the square root, we square both sides of the equation:
  • This transforms the equation into \( r^2 = \frac{3V}{\pi h} \).
  • Squaring is the opposite of square rooting; it cancels out the square root, simplifying the expression.
When working with square roots, remember:
  • Squaring both sides is a useful technique to eliminate the root.
  • Ensure every step is justified to avoid incorrect assumptions.
  • After squaring, check your work by verifying the solution in the original equation if possible.
Mastering square root operations eases the process of solving more complex mathematical expressions by stripping them down to basic components.

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Most popular questions from this chapter

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Use the Pythagorean Theorem and the square root property to solve. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 6 miles long and 3 miles wide. How long is a pedestrian route that runs diagonally across the park?

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is 3.00 dollar. A three-month pass costs 7.50 dollar and reduces the toll to 0.50 dollar. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Find all values of \(x\) satisfying the given conditions. $$ y_{1}=x-1, y_{2}=x+4, \text { and } y_{1} y_{2}=14 $$

Use the Pythagorean Theorem and the square root property to solve. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. The base of a 30 -foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall is the building?
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