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Chapter 1: Problem 99

Use the methods for solving quadratic equations to solve each formula for the indicated variable. \(A=\pi r^{2}\) for \(r\)

Short Answer

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

Step by step solution

01

Understand the equation

The given equation is the formula for the area of a circle: \(A = \pi r^2\). We need to solve this equation for the radius, \(r\).
02

Isolate the squared term

To isolate \(r^2\), divide both sides of the equation by \(\pi\): \[\frac{A}{\pi} = r^2\]
03

Solve for the variable

Take the square root of both sides of the equation to solve for \(r\): \[r = \sqrt{\frac{A}{\pi}}\] Since the radius cannot be negative, we consider only the positive root: \[r = \sqrt{\frac{A}{\pi}}\]

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool for solving equations of the form \(ax^2 + bx + c = 0\). This formula helps us find the values of \(x\) that satisfy the equation. The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. The symbols \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation.
  • \(a\) is the coefficient of \(x^2\).
  • \(b\) is the coefficient of \(x\).
  • \(c\) is the constant term.
To use the formula, substitute the values of \(a\), \(b\), and \(c\) into the equation and solve for \(x\).Remember that \(\pm\) means you will get two solutions, one by adding the square root term and one by subtracting it.
Area of a Circle
The area of a circle is calculated using the formula: \(A = \pi r^2\), where \(A\) stands for the area, and \(r\) is the radius of the circle. This formula tells us how much space is inside the circle. To solve for the radius \(r\), you need to rearrange the equation.
  • First, divide both sides by \(\pi\): \(\frac{A}{\pi} = r^2\).
  • Then, take the square root of both sides: \(r = \sqrt{\frac{A}{\pi}}\).
This gives us the radius in terms of the area. Remember, the radius must be a positive value because distances cannot be negative.
Solving for a Variable
Solving for a variable means isolating the variable on one side of the equation. Let's take the equation \(A = \pi r^2\) and solve for \(r\).
  • First, understand the equation means knowing what each symbol stands for. In this case, \(A\) is the area, and \(r\) is the radius.
  • Next, we isolate the variable \(r\) by dividing both sides by \(\pi\): \(\frac{A}{\pi} = r^2\).
  • Finally, we solve for \(r\) by taking the square root of both sides: \(r = \sqrt{\frac{A}{\pi}}\).
This step-by-step method can be applied to any equation where you need to solve for a specific variable.

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Most popular questions from this chapter

Write an inequality of the form \(|x-a| < k\) or of the form \(|x-a| > k\) so that the inequality has the given solution set. HINT: \(|x-a| < k\) means that \(x\) is less than \(k\) units from \(a\) and \(|x-a|>k\) means that \(x\) is more than \(k\) units from \(a\) on the number line. $$(-\infty,-1) \cup(5, \infty)$$

Weighted Average with Fractions Elizabeth scored \(64,75\) and 80 on three equally weighted tests in French. If the final exam score counts for two- thirds of the grade and the other tests count for one-third, then what range of scores on the final exam would give her a final average over \(70 ?\) All tests have a maximum of 100 points. HINT: For this weighted average multiply the final exam score by \(2 / 3\) and the average of the other test scores by \(1 / 3\).

Final Exam Score Lucky scored 65 points on his Psychology 101 midterm. If the average of his midterm and final must be between 79 and 90 inclusive for a \(B\), then for what range of scores on the final exam would Lucky get a B? Both tests have a maximum of 100 points. Write a compound inequality with his average between 79 and 90 inclusive.

Recall that \(\sqrt{w}\) is a real number only if \(w \geq 0\) and \(1 / w\) is a real mumber only if \(w \neq 0 .\) For what values of \(x\) is each of the following expressions a real mumber? $$\sqrt{5-|x|}$$

Find an exact solution to each problem. If the solution is irrational, then find an approximate solution also. Initial Velocity of a Basketball Player Vince Carter is famous for his high leaps and "hang time," especially while slam-dunking. If Carter leaped for a dunk and reached a peak height at the basket of \(1.07 \mathrm{m},\) what was his upward velocity in meters per second at the moment his feet left the floor? The formula \(v_{1}^{2}=v_{0}^{2}+2 g S\) gives the relationship between final velocity \(v_{1}\), initial velocity \(v_{0}\), acceleration of gravity \(g,\) and his height \(S\). Use \(g=-9.8 \mathrm{m} / \mathrm{sec}^{2}\) and the fact that his final velocity was zero at his peak height. Use \(S=\frac{1}{2} g t^{2}+v_{0} t\) to find the amount of time he was in the air. (figure cannot copy)
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