Measurement and similar triangles grade 10 university level math

Question

I dont know what the point is in the question "Darren marked a horizont cut line on the tree 50cm from the ground". The full word problem is "Darren has to cut down a large tree that has become diseased. To do this safely he must determine the height of the tree and figure out where the top of the tree will land when it falls. Darren marked a horizont cut line on the tree 50cm from the ground. Then Darren cut a stick and held it vertically at arms length infront of him. He moved away from the tree until he could sight the top of the tree over the stick. Darrens reach is .8m long the stick is .7m long he stood 30m away and he is 1.7m tall. PLEASE HELPPPPPP I DONT KNOW HOW TO FIND WHERE IT WILL FALL EITHER. 

Answer 1

Well, at least you know that this problem has to do with similar triangles.  Law 5 of John Milton Gregory’s The Seven Laws of Teaching says, “The truth to be taught must be learned though truth already known.”  Let’s do that.

 

Definition:  Triangles are similarif they have the same shape, but can be different sizes.

 

This definition allows us to set up lots of ratios of the measurements.  When we have one unknown measurement (it is often called x) and all three of the other measurements [a proportion, like in x/a = b/c], we can find x.

 

So, draw a picture!  Label it!  It does not have to be a fancy tree.  Now, identify the two similar triangles:

 

  Triangle 1:  point at Daren’s eye, horizontally the length of Daren’s arm, length of stick Daren is holding – then back to Daren’s eye.

    This triangle is (0.8 m), (0.7 m), and a hypotenuse (since Daren held stick vertically).  We can easily find the length of the hypotenuse using the Pythagorean Theorem.

 

  Triangle 2:  point at Daren’s eye, horizontally all the way to the tree, height of tree above Daren’s eye level – then back to Daren’s eye.

    This triangle is (30 m), (height of tree above Daren’s eye), bigger unknown hypotenuse.

 

Let x = height of tree above Daren’s eye

 

What proportion (equal ratios)do we have [again see your drawing]?

 

       x / (0.7 m)  =  (30 m) / (0.8 m)     [note: PLZ identify these]

 

The easy math:

       x / 0.7 = (30 m) / 0.8          [re-write and cancel one m]

        x = (0.7)(30) / (0.8) m         [multiply both sides by 0.7

        x=26.25 m

 

Oh, yes, since Daren is 1.7 m tall, we add his height (to get measurement from ground)

 

     Distance = 26.25 + 1.7 = 27.95 m

 

And, since the cut Daren made in the tree is 50 cm (or 0.5 m) from the ground, we assume that the falling tree will rotate on that point.  Thus, we subtract that –

 

      Distance = 27.95 – 0.5 = 27.45 m

 

Now, I realize that this is a math problem, but anyone who stands exactly 28 meters from this tree falling is not too smart.