Question 1046158
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There is a bamboo tree 22 feet tall. The upper end of which, being broken, reaches the ground 8 feet from the stem. Find the height of the break?
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Let x be the height of the break.

Then you have an right angled triangle with the legs x ft and 8 ft.

The hypotenuse is {{{sqrt(x^2 + 8^2)}}}.

The height of the bamboo tree is  {{{x + sqrt(x^2 + 8^2)}}}, and it is equal to 22 ft.

Hence, the equation is 

{{{x + sqrt(x^2 + 8^2)}}} = 22.

Simplify:

{{{sqrt(x^2 + 64)}}} = 22 - x.

Square both sides

{{{x^2 + 64}}} = {{{(22-x)^2}}},  or

{{{x^2 + 64}}} = {{{484 - 44x + x^2}}},

64 - 484 = -44x,

44x = 420,

x = {{{420/44}}} = {{{105/11}}} ft.

<U>Answer</U>.  The height of the break is {{{105/11}}} ft = {{{9}}}{{{6/11}}} ft.

<U>Check</U>.  {{{22 - 105/11}}} = {{{137/11}}}.

        {{{(137/11)^2 - (105/11)^2}}} = {{{7744/121}}} = 64,

        {{{sqrt((137/11)^2 - (105/11)^2)}}} = 8.   Correct!
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