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Chapter 1: Problem 120

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

Short Answer

Expert verified
r = \frac{A}{\pi (r + s)}

Step by step solution

01

Isolate the terms with the variable

Given the equation: \[A = \pi r^{2} + \pi r s \]We need to solve for \(r\). Start by isolating the terms that contain \(r\). \[A = \pi r (r + s) \]
02

Factor out the common term

Factor \(\pi r\) out of the right side of the equation: \[A = \pi r (r + s) \]
03

Solve for the variable

To isolate \(r\), divide both sides of the equation by \(\pi (r + s)\): \[r = \frac{A}{\pi (r + s)} \]

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Key Concepts

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

Solving Equations
Equations are mathematical statements that show the equality between two expressions. To solve an equation means to find the value(s) of the variable(s) that make the equation true.
Let's consider the equation given in the exercise: \( A = \pi r^{2} + \pi r s \).
Our goal is to solve for the variable \r \.
Isolating Variables
Isolating the variable involves rearranging the equation so that the variable you are solving for is on one side of the equation by itself.
In our example, we start with: \( A = \pi r^{2} + \pi r s \).
First, identify the terms that include \r \ and factor them out: \( A = \pi r (r + s) \). This equation now has \r \ on one side. To isolate \r \, divide both sides by the term involving \r: \( r = \frac{A}{\pi (r + s)} \).
Now \r \ is isolated!
Factoring
Factoring is a mathematical process where you express a polynomial (a sum of terms) as a product of simpler terms.
It helps simplify equations and solve for unknowns.
In our exercise, we factored out the common term \pi r \ from the right side of the equation: \( A = \pi r (r + s) \).
Factoring made it easier to isolate \r \ and solve the equation. Remember, always look for common factors that can be pulled out to simplify your equation.

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Most popular questions from this chapter

Donovan has offers for two sales jobs. Job A pays a base salary of \(\$ 25,000\) plus a \(10 \%\) commission on sales. Job B pays a base salary of \(\$ 30,000\) plus \(8 \%\) commission on sales. a. How much would Donovan have to sell for the salary from Job A to exceed the salary from Job B? b. If Donovan routinely sells more than \(\$ 500,000\) in merchandise, which job would result in a higher salary?

How is the process to solve a linear inequality different from the process to solve a linear equation?

Explain why \(x^{2}=4\) is equivalent to the equation \(|x|=2\).

A box of cereal is labeled to contain 16 oz. A consumer group takes a sample of 50 boxes and measures the contents of each box. The individual content of each box differs slightly from 16 oz, but by no more than 0.5 oz. a. If \(x\) represents the exact weight of the contents of a box of cereal, write an absolute value inequality that represents an interval in which to estimate \(x\). b. Solve the inequality and interpret the answer.

A die is a six-sided cube with sides labeled with \(1,2,3,4,5,\) or 6 dots. The die is a "fair" die if when rolled, each outcome is equally likely. Therefore, the probability that it lands on " \(1 "\) is \(\frac{1}{6}\). If a fair die is rolled 360 times, we would expect it to land as a "1" roughly 60 times. Let \(x\) represent the number of times a "1" is rolled. The inequality \(\left|\frac{x-60}{\sqrt{50}}\right|<1.96\) gives the "reasonable" range for the number of times that a "1" comes up in 360 rolls. a. Solve the inequality and interpret the answer in the context of this problem. b. If the die is rolled 360 times, and a "1" comes up 30 times, does it appear that the die is a fair die?
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