SOLUTION: Hi, my question is: By what product must the radius of a circle be decreased in order to decrease the area by 75%? Thanks for your help.

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Question 135051: Hi, my question is: By what product must the radius of a circle be decreased in order to decrease the area by 75%?
Thanks for your help.
Answer by ptaylor(2198) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=the product that the radius needs to be decreased
A=pi%2Ar%5E2 Solve for r%5E2
r%5E2=A%2Fpi take square root of each side
r=%2B-sqrt%28A%2Fpi%29 ------------------------eq1
Now, if we decrease the area by 75%, then we have 25% of the area left,so:
%28r-xr%29=%2B-sqrt%280.25A%2Fpi%29 simplifying we have:
%28r-xr%29=%2B-0.5sqrt%28A%2Fpi%29-----------------------------eq2
Now we can see from eq1 that %2B-sqrt%28A%2Fpi%29=r. Substitute this into eq2
r-xr=0.5r subtract r from each side
r-r-xr=0.5r-r collect like terms
-xr=-0.5r divide both sides by r
x=0.5------------this tells us that r has to be decreased by 50% to get a decrease of 75% in the Area
CK
A1=pi%2Ar%5E2----------------eq1
A2=pi%2A%280.5%2Ar%29%5E2 or
A2=pi%2A0.25r%5E2-------------eq2
A2=0.25A1 because in eq2 pi%2Ar%5E2=A1


Hope this helps---ptaylor