Broken Bamboo

Here, AC+BC=32

From triangle, $AC^2$ - $BC^2$ = $AB^2$ = $16^2$

$\rightarrow$ (AC+BC) (AC-BC) = 256

$\rightarrow$ 32 (AC-BC) = 256

$\rightarrow$ AC-BC = 8

From above two equation we can calculate BC = 12

Set BC is x and AC is y, AC=A'C, so A'C =y x=32-y 16squared+xsquared=ysquared (pythagorean theorem) Then use x to find y 256+(32-y)squared=ysquared 256+1024-64y+ysquared=ysquared 256+1024=64y 1280=64y 20=y Then, 32-20=12

$\large \color{#3D99F6}Given$

$\large AB=32 m$

$\large A'B=16 m$

$\large \color{#3D99F6}Let:\space AC=b, \space CB=a$

$\large \color{#624F41}Then: \space b=32-a$

$\large \color{#20A900}{Assumption}:\space$ the bamboo stick is perpendicular on the ground

$\large \color{#3D99F6}Using \space \color{#3D99F6}Pythagoras$

$\large {\color{#D61F06}{b^2=a^2+16^2}}$

$\large {(32-a)^2=a^2+16^2}$

$\large {32^2+a^2-2\times32a=a^2+16^2}$

$\large {2^{10}-2^6a=2^8}$

$\large {\color{#D61F06}a=\Large\frac{2^2 \times 2^8 -2^8}{2^6}}$

$\large {\color{#D61F06}a=\Large\frac{2^8 \times (2^2-1)}{2^6}}$

$\large {\color{#D61F06}a=2^{8-6} \times (2^2-1)=4 \times 3=\boxed{\color{maroon}{12m}}}$

It's pretty easy to guess; since the distance $16$ shown in the diagram is one of the legs of the right triangle, the triangle has to satisfy one of the Pythagorean triples (the answer has to be an integer). In this case, the triplets are $6,8,10$ , multiplying by 2 gives $12,16,20$

The answer is $\boxed{12}$

lets x be the height above the ground,BC=x,AC=(32-x)A'C=(32-x), we have from right angle triangle x^2+16^2=(x-32)^2, by solving the above equation we get x=12

Let the distance from the base to the breaking point be x meter, hence the remaining is (32-x) meter. Therefore, the distance from the tip to the cracking point is equivalent to 32-x. This satisfies the phytagorean theorm of x^2+y^2=z^2. Substituting the hypothenuse with 32-x, adjacent side with x, and the other one with 16, thus we get x =12.

I have figured out a formula. If the height of a bamboo is "h" and the distance between its root and the point at which its vertex touches, when broken, is "h\2". Then the formula for calculating the remaining height is: h(r) = 3h/8.

Let's check it. When h=32, then h(r) = 3*32/8 = 12.

My strategy: Calculating the total area ABA', the subtracting the area shaded blue. Calculating the line CB based on the remaining area.

in pythagorus there is combination as 3,4,5 .in this they are multiply by 4,logical answer 12

The question is asking as to resolve the length of BC

We know that AC = A´C, and from here we can create a couple of equations with two variables each to help us resolve the problem:

• we know that AC + BC = 32;

• according to Pythagoras theorem, AC2 = BC2 + 162 => AC2 - BC2 = 256. This can also be written as: (AC – BC) x (AC + BC) = 256;

We know from the first equation the value of AC + BC, which we can substiture in the second equation: (AC – BC) x 32 = 256 => AC – BC = 256 / 32 => AC – BC = 8

We have now ended up with two equations that are solved easily:

AC + BC = 32 and AC – BC = 8

Which takes us to conclude that BC = 12

Using the 3 - 4 - 5 triangle, we find that 4 x 4 = 16, and thus that the other two are 3 x 4 = 12, and 5 x 4 = 20. Since we need the opposite side of the triangle, we fins that the answer is 12 meters

Relevant wiki: Pythagorean Theorem

Consider the diagram. Using pythagorean theorem , we have

$(32-x)^2=x^2+16^2$

$1024-64x+x^2=x^2+256$

$768=64x$

$\boxed{x=12}$

From the problem:

$BC+AC=32$

We can solve for $AC$ :

$AC=32-BC$

From the Pythagorean Theorem:

$AB^2+BC^2=AC^2$

Substituting in data from the problem we get:

$16^2+BC^2=AC^2$

$\therefore256+BC^2=AC^2$

Now we have two ways to express $AC^2$

$AC^2=(32-BC)^2$

$AC^2=BC^2+256$

so we can set those equal to each other and solve for $BC$ .

$(32-BC)^2=BC^2+256$

$\therefore 1024-64BC+BC^2=BC^2+256$

$\therefore 1024-64BC=256$

$\therefore -64BC-768$

$\therefore BC=12$

The no of squares in side cb = 5x The no of squares in side c a' = 8x By pathagorus Therefore, CB equal 12 cm

I took the clever way out. The black lines dividing up the piece of bamboo look more or less even. So I counted them up (14 total). 32/14=2.28 .. 2.28 * 5 = 11.39 .. Round up .. Boom 12. ;)