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Chapter 11: Problem 1

Solve for the indicated variable. \(A=\pi r^{2}\) for \(r\)

Short Answer

Expert verified
To solve the equation \(A=\pi r^2\) for the variable \(r\), follow these steps: 1. Divide both sides by \(\pi\): \(\frac{A}{\pi} = r^2\) 2. Take the square root of both sides: \(r = \sqrt{\frac{A}{\pi}}\)

Step by step solution

01

Divide Both Sides by \(\pi\)

In order to isolate \(r^2\) on one side of the equation, we will divide both sides by \(\pi\). \(A = \pi r^2\) \(\frac{A}{\pi} = \frac{\pi r^2}{\pi}\) The \(\pi\) cancels out on the right side: \(\frac{A}{\pi} = r^2\)
02

Take the Square Root of Both Sides

Now that we have an equation with \(r^2\) on one side, we can take the square root of both sides to isolate \(r\). \(\sqrt{\frac{A}{\pi}} = \sqrt{r^2}\) The square root of \(r^2\) is \(r\), so our final equation is:
03

The Solution

We have found that, to solve for \(r\) in the equation \(A = \pi r^2\), use the formula: \(r = \sqrt{\frac{A}{\pi}}\)

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Key Concepts

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

Solving for a Variable
When solving for a specific variable in an equation, our goal is to isolate that variable on one side of the equation. This process involves algebraic manipulation, where we use various algebraic operations to rearrange the equation. For instance, in the given exercise, we needed to solve for the variable \( r \) in the equation \( A = \pi r^2 \). Here鈥檚 how we approach it:

  • Identify the term containing the variable: In our equation, \( \pi r^2 \) is the term with the variable \( r \).
  • Perform operations to isolate the variable: We begin by eliminating \( \pi \) from \( \pi r^2 \) through division. This leaves us with \( \frac{A}{\pi} = r^2 \).

Rewriting equations to solve for a variable is a fundamental skill in algebra. It applies to many different types of equations in mathematics and science. By consistently practicing these steps, you become proficient in rearranging formulas and making calculations easier.
Area of a Circle
The area of a circle is a well-known formula in geometry given as \( A = \pi r^2 \), where:
  • \( A \) represents the area of the circle.
  • \( \pi \) is a constant approximately equal to 3.14159.
  • \( r \) is the radius of the circle, which is the distance from the center of the circle to any point on its boundary.

Understanding this formula is essential for solving geometric problems involving circles. Here, it helped us find \( r \) by manipulating the formula. Note that the formula arises from the properties of the circle and the concept of \( \pi \), which defines the ratio of the circumference of a circle to its diameter.

Learning how to use the area formula efficiently can assist with various applications, such as finding the amount of material needed to cover a circular region or calculating the space within a circular boundary.
Square Root Property
The square root property is a useful tool in algebra for solving equations where a variable is squared. Simply put, the property states that if \( x^2 = a \), then the solutions for \( x \) are \( x = \sqrt{a} \) and \( x = -\sqrt{a} \).

In our original problem, when we arrived at \( \frac{A}{\pi} = r^2 \), we needed to apply the square root property to solve for \( r \). By taking the square root of both sides, we get:

  • \( r = \sqrt{\frac{A}{\pi}} \)

This illustrates a key practice in algebra鈥攗sing inverse operations to solve for the unknown. Applying the square root property is a practical step, especially in quadratic equations, to transition from a squared term to a linear variable.

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Most popular questions from this chapter

Multiply and simplify. $$\frac{1}{2}(18+7 i)$$

Attendance at Broadway plays from \(1996-2000\) can be modeled by $$ y=-0.25 x^{2}+1.5 x+9.5 $$ where \(x\) represents the number of years after 1996 and \(y\) represents the number of people who attended a Broadway play (in millions). a) Approximately how many people saw a Broadway play in \(1996 ?\) b) In what year did approximately 11.75 million people see a Broadway play?

Solve. \((5 a-3)^{2}+6(5 a-3)=-5\)

SOLVE. \(63=7 x^{2}\)

Divide. Write the result in the form \(a+b i\). $\frac{2 i}{-3+7 i}$$
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