SOLUTION: if the radius of a circle is increase by 4 units, its original area is multiplied by 2. Find the original radius.

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Question 201095: if the radius of a circle is increase by 4 units, its original area is multiplied by 2. Find the original radius.
Found 2 solutions by Earlsdon, Alan3354:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Try this:
Original area is:
A%5B1%5D+=+pi%2Ar%5E2
The area of the new circle is:
A%5B2%5D+=+2%2AA%5B1%5D and A%5B2%5D+=+pi%2A%28r%2B4%29%5E2, so...
pi%2A%28r%2B4%29%5E2+=+2%2Api%2Ar%5E2
pi%2A%28r%5E2%2B8r%2B16%29+=+2%2Api%2Ar%5E2 Divide both sides by pi
r%5E2%2B8r%2B16+=+2%2Ar%5E2 Subtract r%5E2 from both sides.
8r%2B16+=+r%5E2 Rewrite as a quadratic equation in standard form:
r%5E2-8r-16=0 Solve using the quadratic formula: r+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a where: a = 1, b = -8, and c = -16.
r+=+%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%281%29%28-16%29%29%29%2F2%281%29 Simplify:
r+=+%288%2B-sqrt%2864-%28-64%29%29%29%2F2
r+=+%288%2B-sqrt%28128%29%29%2F2
r+=+4%2B4sqrt%282%29 or r+=+4-4sqrt%282%29 or approximately...
highlight_green%28r+=+9.65685%29 or cross%28r+=+-1.65685%29 Discard the negative solution as the radius can only be a positive value!

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
if the radius of a circle is increase by 4 units, its original area is multiplied by 2. Find the original radius.
-------------------
A+=+pi%2Ar%5E2
2A+=+2pi%2Ar%5E2
2A+=+pi%2A%28r%2B4%29%5E2
2r%5E2+=+%28r%5E2+%2B+8r+%2B+16%29
2r%5E2+=+r%5E2+%2B+8r+%2B+16
1r%5E2+-+8r+-+16+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B-16+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-8%29%5E2-4%2A1%2A-16=128.

Discriminant d=128 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+128+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+128+%29%29%2F2%5C1+=+9.65685424949238
x%5B2%5D+=+%28-%28-8%29-sqrt%28+128+%29%29%2F2%5C1+=+-1.65685424949238

Quadratic expression 1x%5E2%2B-8x%2B-16 can be factored:
1x%5E2%2B-8x%2B-16+=+%28x-9.65685424949238%29%2A%28x--1.65685424949238%29
Again, the answer is: 9.65685424949238, -1.65685424949238. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B-16+%29

Sub r for x and ignore the negative answer.
r = 4 + 4sqrt(2)
r = ~ 9.65685