word problem

Question

Buried treasure. Ahmed has half of a treasure map,which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x?

Answer 1

In order to solve this problem, start by drawing a triangle. From here we know that Ahmed's map says the treasure is buried 2x+6 paces from Castle Rock. This shows that the treasure is buried in a straight line from Castle Rock 2x+6 paces so this will represent the Hypotenuse of the triangle.

 

Now lets look at Vanessa's map, which says to walk x paces North of Castle Rock and then 2x+4 to the East. Therefore draw a vertical line to represent the paces North (x) and then a line from the top of the previous line to the right to represent paces to the East (2x+4). These represent the adjacent and opposite sides of the triangle you made earlier. So you should now have a complete triangle, the adjeacnt and opposite sides represented by Vanessa's map and the hypotenuse represented by Ahmed's map.

 

Now you can fill in length of these sides with the equations given to you in the problem.

 

If you can remember anything about right angle triangles, it's pythagoreans theorem. (Otherwise known as a^2 + b^2 = c^2) a, b, and c represent the three sides of the triangle where a and b are the adjacent and opposite sides and c is the hypotenuse. You can now substitute in the equations from the problem for a, b, and c. Your equation should be:

 

(x)^2 + (2x+4)^2 = (2x+6)^2

 

From here things get tricky and require some algebra. In an effort not not give too much away and answering the problem for you I will leave the rest for you. However, I would suggest to start by breaking the squared equations into two seperate equations and then working the equation out. An example of this is as follows:

 

(10x+2)^2 = (10x+2)(10x+2) = 100x^2 + 40x + 4

 

From this you can balance the equation to form an equation equal to zero. Here's an example:

 

2x^2 + 5x + 26 = x^2 - 5x + 2       =>         x^2 + 10x + 24 = 0

 

And lastly simplify this equation to find two possible x values as shown next using the example equation from above:

 

x^2 + 10x + 24 = 0       =>       (x+6)(x+4)

 

If this was our problem, there would be two possible answers. Your values will be inverse the values from the equations so despite the equations having +6 and +4, x can be either -6 or -4. If they were negative values in the equation then x would be positive.

 

Now I know it doesnt sound possible having two answers but most likely you will have one value that is positive and one which is negative. Since this question deals with the real world your answer should be the positive and more reasonable value.

 

Once you get an x value try it out, plug it back into your original equations and get a length for each side of the triangle, then apply pythagoreans theorem. If it works then you have sucessfully answered the problem.

 

Feel free to ask any questions you may have, I know there are some tricky steps. Also if you want post the answer you come up with and I can check it for you.

 

Good Luck!

Answer 2

 Information given look like:

 

 A right triangle with Pythagoras theorem:

 

 (2X + 6 ) = X ^2 +( 2X +4 )^2

 

 

 4X^2 + 24X + 36 = X^2 + 4X^2 +16X +16

 

      X^2 -8 X -20 = 0

 

      ( X - 10 ) ( X +2 ) =0

 

      X = 10    X = -2

 

     X=10