A Challenging Math Problem, see if you can answer it, cuz i couldnt? Explain how you got it too.?

This famous puzzle is about a water lily that Henry W. Longfellow introduced into his novel, Kavenaugh:

When the stem of a water lily is vertical the blossom is 10 centimeters above the surface of the lake. If you pull the lily to one side keeping the stem straight the blossom touches the water at a spot 21 centimeters from where the stem formerly cut the surface. How deep is the water?

A Challenging Math Problem, see if you can answer it, cuz i couldnt? Explain how you got it too.?
stem seen = 10

stem underwater = x = lake's depth

actual stem's length = x+10



blossom's radius = r

now more y cm being submerge.

only 10-y can be seen when blossom touching water.

x+y is underwater.



21 cm is line ABC, A where stem cut surface BEFORE, B where stem cut surface AFTER, C where blossom touch water. AB=a and BC=b,

a+b=21, b=21-a



b^2 = r^2 + (10-y)^2

=(21-a)^2

(x+y)^2 = x^2 + a^2

2xy+y^2 = a^2



a^2 -42a +441 = r^2 +y^2 - 20y +100

2xy -42a + 441 = r^2 -20y +100



r/x = (10-y)/a = (21-a)/(x+y)



(10-y)(x+y) = a(21-a)

10x+10y-xy-y^2 = 21a-a^2

10x+10y+xy = 21a

20x+20y+2xy = 42a

2xy -42a + 441 = r^2 -20y +100

341 = r^2 +20x

r=0,

x = 341/20 = 17.05cm
Reply:this requires the use of muliple triangles.



First develop your right triangle that is above the water.

10, 21 and c

10^2 + 21 ^2 = c^2 c=23.26



Now you know the length of the base of the isoscoles triangle with the verticies of the bottom of the water, the place where the lily touches the water, and the original position of the lily.



Find the angle of the vertex from the original position to where it touches the water.



arctan(21/10) = 64.5 degrees



Bisect the 51 degree angle to bisect the base of the triangle.

1/2 base = .5 * 23.26 = 11.63 cm



11.63 / cos(64.5) = length of stem = 27.01



Subtract 10 cm from 27.01 to figure out how deep the water is.



27.01 - 10 = 17.01 cm

= = = = =

Another solution:

The three sides of the right triangle are x, 21 and x+10.

a: x: depth of water

b: 21 distance from standing lily and the point it touches the water.

c: x+10: length of lily stem



a^2 + b^2 = c^2

x^2 + 21 ^2 = (x + 10)^2

341 / 20 = x = 17.05 cm is the water depth



I guess it all depends on how much trig and geometry you remember. :)
Reply:you can construct a rt angle triangle with the information given



let x = depth of water



then the 3 sides of the triangle are



x, x+10, and 21 where x+10 is the hypotenuse



using Pythagorean theorem



x^2 + 21^2 = (x+10)^2



solve for x



x = 8.98 cm
Reply:Easy: 10cm less than the length of the water lily's stem.



Actually the problem is not challenging at all, you just haven't drawn a diagram of what you know.

1) Find the angle from the start to the ending position. 2) Find the linear distance from the start to the end. 3) find the length of the lily stem.