/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 95 Find the center and radius of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 95

Find the center and radius of the circle, and sketch its graph. \(\left(x-\frac{1}{2}\right)^2+\left(y-\frac{1}{2}\right)^2=\frac{9}{4}\)

Short Answer

Expert verified
The center of the circle is at \((\frac{1}{2}, \frac{1}{2})\) and the radius is \(\frac{3}{2}\).

Step by step solution

01

Identify center and radius

Look at the given equation \(\left(x-\frac{1}{2}\right)^2 + \left(y-\frac{1}{2}\right)^2 = \frac{9}{4}\). From this, we see that coordinates of the center (h, k) are \(\frac{1}{2}\) and \(\frac{1}{2}\), and the radius \(r\) is sqrt(\(\frac{9}{4}\)), which simplifies to \(\frac{3}{2}\).
02

Plot Center

On a graph, plot the center of the circle at the point (\(\frac{1}{2}\), \(\frac{1}{2}\)). This will be the mid-point of your circle.
03

Draw Circle

Using the value of radius as \(\frac{3}{2}\), draw a circle around the center such that all points of the circle are \(\frac{3}{2}\) units far from the center. This can be done using a compass if you're drawing by hand, or using a circle drawing tool if you're using a digital tool.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Center of a Circle
The center of a circle is a key concept in geometry that defines the middle point from which every point on the circle is equidistant. When reading the standard form of a circle鈥檚 equation, \[(x-h)^2 + (y-k)^2 = r^2,\]we identify the center as the point \((h, k)\). In our exercise, the equation is given as:\[\left(x-\frac{1}{2}\right)^2 + \left(y-\frac{1}{2}\right)^2 = \frac{9}{4}.\] From here, we clearly see that \(h\) and \(k\) are both \(\frac{1}{2}\).
  • The center is at \((\frac{1}{2}, \frac{1}{2})\).
  • This point represents the middle of the circle.
  • All points on the circle are equally distant from this center.
Understanding the center helps you determine how the circle is positioned on a graph.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle. It plays a crucial role in defining the size of the circle. In the standard circle equation, \(r^2\) is found on the right side. We have:\[\left(x-\frac{1}{2}\right)^2 + \left(y-\frac{1}{2}\right)^2 = \frac{9}{4}.\] To find the radius, take the square root of \(\frac{9}{4}\): \[\sqrt{\frac{9}{4}} = \frac{3}{2}.\]
  • The radius is \(\frac{3}{2}\), or 1.5 units.
  • It tells us how large the circle is.
  • Every point on the circumference is 1.5 units away from the center at \((\frac{1}{2}, \frac{1}{2})\).
Knowing the radius is essential for graphing the circle accurately.
Graphing Circles
Graphing a circle involves accurately plotting its center and using the radius to draw the circle. Begin by identifying the center \((h, k)\). In our case, it is \((\frac{1}{2}, \frac{1}{2})\). Mark this point carefully.
  • Initial Step: Locate \((\frac{1}{2}, \frac{1}{2})\) on your graph.
  • Next: Use the radius \(\frac{3}{2}\) to measure out from the center to every point on the circle.
A compass or digital tool can help you maintain accuracy.
  • Ensure that the circle is smooth and centered around \((\frac{1}{2}, \frac{1}{2})\).
  • All edge points should be exactly \(\frac{3}{2}\) units away from the center.
Graphing accurately ensures that the circle reflects the equation correctly on a coordinate plane.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt{x} $$

The function given by $$ f(x)=k\left(2-x-x^{3}\right) $$ has an inverse function, and \(f^{-1}(3)=-2\). Find \(k\).

College Students The numbers of foreign students \(F\) (in thousands) enrolled in colleges in the United States from 1992 to 2002 can be approximated by the model. $$ F=0.004 t^{4}+0.46 t^{2}+431.6, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 1992. (Source: Institute of International Education) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least.

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ (f \circ g)^{-1} $$

Cost, Revenue, and Profit A company produces a product for which the variable cost is \(\$ 12.30\) per unit and the fixed costs are \(\$ 98,000\). The product sells for \(\$ 17.98\). Let \(x\) be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Write the profit \(P\) as a function of the number of units sold. (Note: \(P=R-C\) )
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.