A tree casts a shadow 24 feet long. At the same time, a vertical rod 16 feet high casts a shadow 6 feet long. How tall is the tree?

Question

would like help with this question can't seem to figure it out please and thank you! Would be greatly appreciated 

Answer 1

It helps to draw a picture.

 

For our picture, the Sun is at a fixed position.  Draw sunlight to a fixed point on a flat ground.  The tree is at the point 24 feet from this fixed point (thus the base of this right triangle is 24 feet).

 

The vertical rod is at the point on the sun ray where the height is 16 feet.  The rod's shadow is 6 feet long.  If we wanted to, we could use the Pythagorean Theorem to calculate the hypotenuse of this triangle, but that's not the question.

 

This is a proportion question.  Any of the proportional relationships may be set equal to each other

   for example:

       the height of the rod is to the height of the tree is as the shadow of the rod is to the shadow or the tree

       the height of the rod is to the shadow of the rod as is the height of the tree is to the shadow of the tree

       . . .

Note:  these are two fractions set equal and they may be turned upside down, cross-multiplied, etc as the rules of algebra allow.

 

Let's keep T (the height of the tree in the numerator because the question asked for that and because I think it makes the math easier). 

    So, the height of the tree is to the height of the rod is as the shadow of the tree is to the shadow of the rod.                 

        T / 16  =  24 / 6

          T =    4*16         (simplify 24/6 and multiply both sides by 16)

          T = 64 feet

Answer 2

First draw a line for the flat ground. Next, draw the tree, the rod and the lines indicating the sunlight's path down to the ground. The distance from the tree (or rod) to the point where the sunlight's path intersects the ground is the shadow. 

 

This drawing should create two similar right triangles. 
Since the triangles are similar, we can set up a proportion and solve.

 

  tree height / shadow length  :  X/24 = 16/6 

 

6 × X = 24 × 16

6X = 384

to solve for X, we do inverse operations and divide each side by 6.

X = 64 

 

Looking back at the problem, we defined X as the height of the tree. The units for this number are feet so, we can conclude that the tree is 64 feet tall. 

Answer 3

Use a proportion:

 

tree / shadow = rod / shadow

T / 24 = 16 / 6                         Multiply both sides by 24

T = (24)(16) / 6

T = 384 / 6

T = 64

 

The tree is 64 feet tall.