SOLUTION: 146. Height of a post. Betty observed that the lamppost in the front of her house cases a show of length 8 feet when the angle of inclination of the sun is 60 degrees. How tall is

Algebra ->  Triangles -> SOLUTION: 146. Height of a post. Betty observed that the lamppost in the front of her house cases a show of length 8 feet when the angle of inclination of the sun is 60 degrees. How tall is      Log On


   

Question 58585: 146. Height of a post. Betty observed that the lamppost in the front of her house cases a show of length 8 feet when the angle of inclination of the sun is 60 degrees. How tall is the lamppost? (In a 30-60-90 right triangle, the side opposite 30 is one-half the length of the hypotenuse)
not sure how to solve
Found 3 solutions by Earlsdon, stanbon, Edwin McCravy:
Answer by Earlsdon(6294) About Me  (Show Source):
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You can use the Pythagorean theorem to solve this problem: c%5E2+=+a%5E2+%2B+b%5E2 Where: c = the length of the hypotenuse and a & b are the lengths of the other two sides of the right triangle.
Let's use a for the height of the lamp post. And, as you state in your problem, in a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse. So you can write:
c = 2(8 ft) = 16 ft.
%2816%29%5E2+=+8%5E2+%2B+a%5E2 Simplify and solve for a, the height of the lamp post.
256+-+64+=+a%5E2
192+=+a%5E2 Take the square root of both sides.
13.86+=+a
The lamp post is 13.86 feet high (approximately)

Answer by stanbon(75887) About Me  (Show Source):
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Betty observed that the lamppost in the front of her house cases a shadow of length 8 feet when the angle of inclination of the sun is 60 degrees. How tall is the lamppost? (In a 30-60-90 right triangle, the side opposite 30 is one-half the length of the hypotenuse)
not sure how to solve
-----------
Draw the picture.
You should see a right triangle with the right
angle at the base of the pole and the 60 degree
angle opposite the pole at ground-level.
The distance from the vertex of the 60 degree
angle to the base of the pole is the shadow
which is 8 ft long.
The angle at the top of the pole is 30 degrees
and the shadow is the side opposite the 30 degree
angle.
So, sine the side opposite the 30 degree angle
is one-half the hypotenuse, the hypotenuse must
be 16 ft.
Now, using Pythagoras: hypotenuse^2 = 8^2 + pole^2
16^2 = 8^2 + pole^2
256 = 64 + pole^2
pole^2 = 192
pole^2 = 64*3
pole = 8sqrt(3)
Cheers,
Stan H.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Height of a post. Betty observed that the lamppost in the front of her house
cases a show of length 8 feet when the angle of inclination of the sun is 60
degrees. How tall is the lamppost? (In a 30-60-90 right triangle, the side
opposite 30 is one-half the length of the hypotenuse) 
not sure how to solve 

                |
              - O -
                |


                       
          |
        --o--
         /|
        / | 
     h /  | 
      /30°|x
     /    | 
    /     | 
   /60°   |
   ¯¯¯¯¯¯¯
     8 ft

Let x be the height of the lamp post. I have marked
the hypotenuse h. The shadow is the line marked 8 ft. 
It is the side opposite the 30° angle. Therefore 

h = 2 × 8 ft or 16 ft. 

I will now re-draw the picture putting
16 ft. where the h is:

                |
              - O -
                |


                       
          |
        --o--
         /|
        / | 
    16 /  | 
      /30°|x
     /    | 
    /     | 
   /60°   |
   ¯¯¯¯¯¯¯
     8 ft


Now we can use the Pythagorean theorem

       c² = a² + b²

where c = 16, a = 8, and b = x. Substitute these

       c² = a² + b²
    (16)² = (8)² + x²
      256 = 64 + x²

Subtract 64 from both sides

      192 = x²

Take square root of both sides:
      ___
     Ö192 = x
     13.9 = x

The lampost is approximately 13.9 ft tall.

Edwin