Sine laws with Oblique Triangles: The Tower of Pisa

[HOFBrINCl]

Here's the question: The leaning Tower of Pisa leans toward the south at an angle of 5.5°. On one day, its shadow was 90m long, and the angle of elevation from the tip of the shadow to the top of the tower is 32°.

Determine the slant height of the tower.

How high is the tip of the tower above the ground?

Now what I couldn't get is what is the "slant height?" I did the equation using the sine law, but then realized I had just found the length of the slant, not the height. Any suggestions?

 

[konthelion]

The "slant height" of a triangle is the height of one of its sloping sides, in this case the side where the leaning tower of pizza is leaning towards.

So, yes the length of the slant is the "slant height". The height of the leaning tower would be different (which was what you thought it was).

 

[Architeuthis Dux]

I would suggest using the Pythagorean theorem to find the height of the tower. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the slant height) is equal to the sum of the squares of the other two sides. In this case, the slant height is the hypotenuse, the shadow length is one side, and the height of the tower is the other side.

To find the slant height of the tower, we can use the sine law as you mentioned. Using the given angle of elevation and the shadow length, we can find the length of the slant. However, to find the height of the tower, we need to use the Pythagorean theorem.

So, to find the slant height, we can use the formula:

sin(32°) = slant height/90m

Solving for the slant height, we get slant height = 90m * sin(32°) = 46.3m.

Now, to find the height of the tower, we can use the Pythagorean theorem:

height of tower^2 + 90m^2 = (46.3m)^2

Solving for the height of the tower, we get height of tower = √[(46.3m)^2 - (90m)^2] = 42.6m.

Therefore, the height of the tower is approximately 42.6m above the ground.

In summary, the slant height is the length of the hypotenuse in a right triangle, while the height is the length of the side opposite the right angle. It is important to carefully consider the given information and the appropriate formulas to use in order to find the correct solution.