Your Algebra Homework Can Now Be Easier Than Ever!

Math Problem of the Week 11

Solution by the organizers . This solution does not require the concept of curvature .
We first find the largest circle that touches the parabola at (0, 0). The center
of such disk must be located at (0, r) where r is the radius of the circle. Then the
equation of this circle is

To find the largest possible r we require that this circle intersects the parabola y = x2
at a single point. Solving the system y = x2 and x2 + (y − r)2 = r2 we get

Thus we get two solution , y = 0 and y = 2r −1. However, y needs to be non- negative
since y = x2, thus to have only one solution we need 2r − 1 ≤ 0. That is r ≤ 1/2.
Thus r = 1/2 is the largest radius of a circle touching the parabola at (0, 0).

Now, let be a point on the parabola. By symmetry, the largest
circle C on the concave part of the parabola touching the parabola at must
also touch the parabola at , and thus have a center on the y-axis. Suppose
that the center of such circle is (0, a). The slope of the tangent line to the parabola ,
at , equals the derivative y' = 2x evaluated at x0. Thus the slope of that line
is 2x0. Since the segment joining (0, a) and is perpendicular to the tangent
line at , we have that

Thus . Let r be the radius of C, then we have that .
Plugging the value of a we get . Therefore r > 1/2. Thus the
circle of radius 1/2 will freely rotate on the parabola, and 1/2 is the radius of the
largest such possible circle.
y = x2

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of April 24th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2018, Algebra-Answer.Com.  All rights reserved.