Question 1201828
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water is flowing through a pipe with radius 14 cm.The maximum depth of the water is 9 cm.
What is the width, PQ, of the surface of the water?
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        The solution by @math_tutor2020 is  INCORRECT.

        The error is in incorrect using the intersecting chords theorem.

        I came to bring a correct solution.

        My solution is in two different forms,  for better clarity.



<pre>
                    Solution 1


In the post by math_tutor2020, you can see the right-angled triangle 
with the hypotenuse of 14 cm (the radius from the center to the point P or Q)
and one leg of 14-9 = 5 cm.

Hence, half of PQ is  {{{sqrt(14^2-5^2)}}} = {{{sqrt(196-25)}}} = {{{sqrt(171)}}}.


Then the segment PQ itself is  {{{2*sqrt(171)}}} = 26.1534 cm  (rounded)   <U>ANSWER</U>



                    Solution 2


You can apply the intersecting chords theorem - but you should to use it in correct way.

Then two parts of the horizontal chord are x cm each,
while two parts of the vertical chord are 9 cm and 5+14 = 19 cm.


The intersecting chords theorem takes the form

    x*x = 9*19,  or  x^2 = 171,  x = {{{sqrt(171)}}},  PQ = 2x = {{{2*sqrt(171)}}} = 26.1534 cm  (rounded)   <U>ANSWER</U>


giving the same number.
</pre>

Solved.


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On the intersecting chords theorem, &nbsp;see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-parts-of-chords-intersecting-inside-a-circle.lesson>The parts of chords that intersect inside a circle</A>, 

in this site.