what is the height,x, of the tree?

Question

At a certain time of day, a tree that is x feet tall casts a shadow that is x − 21 meters long. If the distance from the top of the tree to the end of the shadow is x + 6 meters, what is the height, x, of the tree?

Answer 1

I believe that the height is mis-stated here in terms of the units. It reads x feet tall, but I believe it should say x meters. Either that or the other quantities should be in feet. Since 2 of the 3 distances were given in meters, I made the assumption that everything is represented in meters as well. 

 

Now, I used the Pyth. Theorem with X + 6 as the hypotenuse, since it is obviously longer than either X or x - 21.

 

so, x^2 + (x - 21) ^2 = (x + 6) ^2  which yields   x ^2 + x^2 - 42x + 441 = x^2 + 12x + 36  when expanded

 

by subtracting (on both sides) by all terms located on the right side of the equation and combining like terms we obtain

 

x^2 -54x + 405 = 0.    We can factor the left side to obtain (x - 45) (x - 9) = 0 which means x can be either 9 or 45.

 

The answer x = 9 must be discarded since the shadow is 21 meters less, which would make it  negative 12 meters (which is nonsense).

 

Therefore, the height is 45 meters. The shadow would be 24 meters while the distance from the top of the tree to the end of the shadow would be 51 meters.

 

To check the solution, verify that 45^2 + 24^2 = 51^2.

Now, 45^2 = 2025; 24^2 = 576; 2025 + 576 = 2601; 51 ^2 = 2601 as well. CHECK !!