Please help me with this factoring problem !

Question

At a certain time of day, a tree that is x meters tall casts a shadow that is x − 17 meters long. If the distance from the top of the tree to the end of the shadow is x + 8 meters, what is the height, x, of the tree?

Answer 1

The tree and the shadow make a right triangle, so we can use the pythagorean theorem for the sides, with X + 8 as the hypotenuse.

 

X^2 + (x-17)^2 = (x+8)^2

 

After you remove the parenthesis and add like terms, you need to put everything on one side and make it equal to zero.  You should have

 

x^2 - 50x + 225 = 0

 

The next step is to solve the equation by completing the square

 

X^2 - 50x = -225

X^2 -50x + 25^2 = 25^2 - 225

(X-25)^2 = 625 - 225

(x-25)^2 = 400

 

Now take the square of both sides

 

x - 25 = +/- 20

x = 25 - 20

x = 25 + 20

 

x = 45 or x = 5 is the answer.  Then substitute your values of x and see the answer is x = 45 because x - 17 is a negative number, and it doesn't make sense to have a negative number for the shadow.

 

You can test your answer by substituting 45 for x in the equation.

Answer 2

Dear Nicole,

     This problem is a Pythagorean problem.  You must set it up before you can factor it.  Here are some steps to follow:

Question:  Find "X," which is the height of the tree.

1.  Draw the model and label the parts.  You should end up with a triangle. 

 

The height of tree=X meters,  Base of triangle=shadow= X-17,  Hypotenuse= X+8.

 

2.  Use the Pythagorean Equation, which is a²+b²=c² , where "c" is the hypotenuse of the triangle and "b" and "C" are its legs.

 

Use substitution:

3.  (X-17)² +(X)² = (X+8)²     Multiply it out:  (X-17)(X-17) +X²=  (X+8)(X+8)

 

4.  X² +X² -34X+289= X² +16X +64    In order to factor, you need to get everything on the left side and zero on the right side of the equation.

 

5.  X² +X²-X² -34X-16X+289-64=0=Y                    Collect like terms.

 

6.  X²-50X +225=0                                  Now, you may factor the equation.

 

7.  (X -45) (X-5 )  = 0                   

 

There are two solutions.  However, if you substitute one solution at a time into the equation on number 3, only one of them works. Is it X=45 or X=5?

 

      X-45=0   or  X-5=0

         X=45       or X=5                                  

 

Answer:____X=_______

 

I am going to let you finish the problem.  If I helped you, please give me a thumbs up.

                                                                       Susan C.