November 9th, 2017, 02:42 PM  #1 
Newbie Joined: Nov 2017 From: Canada Posts: 4 Thanks: 0  Related rates  inscribed
A rectangle is inscribed in a semicircle of a radius 5m. Estimate the increase in the area of the rectangle using differentials if the length of its base along the diameter is increased from 6m to 6 1/6 m ?

November 9th, 2017, 07:39 PM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,898 Thanks: 716 
Show your work you lazy Canuck!! 
November 10th, 2017, 04:15 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,821 Thanks: 750 
A rectangle "inscribed in a circle" can't have its base "along the diameter". Your question makes no sense.

November 10th, 2017, 04:45 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,145 Thanks: 1418 
The problem states "inscribed in a semicircle", not "inscribed in a circle", so it does make sense.

November 10th, 2017, 04:56 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,821 Thanks: 750 
Right. Thanks, I misread it. Igorrrawr, set up a coordinate system with the origin at the center of the semicircle, the xaxis along its base. We can write the semicircle as $y= \sqrt{25 x^2}$. The vertices of the rectangle on the semicircle are of the form $\left(x, \sqrt{25 x^2}\right)$ and $\left(x, \sqrt{25 x^2}\right)$ for some positive x. The area is given by $A= 2x\sqrt{25 x^2}$. Initially, the base is 6 so x= 3 and $A= 6\sqrt{25 9}= 6(4)= 24$. Differentiate the formula for A to find dA in terms of x and dx, then set dx= 1/6. 

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