SOLUTION: I'm not sure on how to start the following: When mineral deposits formed a coating 1mm thick on the inside of a pipe, the area through which the fluid can flow was reduced by 20%.

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Question 174611This question is from textbook Algebra and Trigonometry Structure and Method Book 2
: I'm not sure on how to start the following:
When mineral deposits formed a coating 1mm thick on the inside of a pipe, the area through which the fluid can flow was reduced by 20%. Find the original inside diameter of the pipe.
It also gives me the equations for area of a circle and diameter.
Help is appreciated.
Thank you. This question is from textbook Algebra and Trigonometry Structure and Method Book 2

Found 2 solutions by checkley77, Alan3354:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for the area of a circle is pir^2
Thus:
.8pir^2=pi(r-1)^2
.8*3.14r^2=3.14(r^2-2r+1)
2.514r^2=3.14r^2-6.28r+3.14
3.14r^2-2.514r^2-6.28r+3.14=0
.626r^2-6.28r+3.14=0
Using the quadratic equation we get 9.504 mm. for the original inside radius of the pipe.
Proof:
.8*3.14*9.504^2=3.14*(9.504-1)^2
2.512*90.326=3.14*8.504^2
226.9=3.14*72.318
227~227

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
When mineral deposits formed a coating 1mm thick on the inside of a pipe, the area through which the fluid can flow was reduced by 20%. Find the original inside diameter of the pipe.
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Reduced by 20% --> 80% remaining, or 0.8 of the original area.
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0.8pi%2Ar%5E2=pi%28r-1%29%5E2
Divide by pi
0.8r%5E2+=+r%5E2-2r%2B1
0.2r%5E2+-+2r+%2B+1+=+0
r%5E2+-+10r+%2B+5+=+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-10x%2B5+=+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=%28-10%29%5E2-4%2A1%2A5=80.

Discriminant d=80 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+80+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+80+%29%29%2F2%5C1+=+9.47213595499958
x%5B2%5D+=+%28-%28-10%29-sqrt%28+80+%29%29%2F2%5C1+=+0.52786404500042

Quadratic expression 1x%5E2%2B-10x%2B5 can be factored:
1x%5E2%2B-10x%2B5+=+%28x-9.47213595499958%29%2A%28x-0.52786404500042%29
Again, the answer is: 9.47213595499958, 0.52786404500042. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B5+%29

Sub r for x, the solver always uses x.
The second answer is less than 1, so it is extraneous. Ignore it.
r+=+5%2B2sqrt%285%29+mms
r = apx 9.472 mms