# The Leaning Tower of Pisa leans toward the south at an angle of 5.5°. One day near noon its shadow was measured to be 84.02 m long and the angle of elevation from the tip of the shadow to the top...

The Leaning Tower of Pisa leans toward the south at an angle of 5.5°. One day near noon its shadow was measured to be 84.02 m long and the angle of elevation from the tip of the shadow to the top of the tower at that time was measured as 32.0°. To answer the following, assume that the Tower is like a pole stuck in the ground, that is, it has negligible width. (Also it is important to know that Pisa Italy is at a latitude of approximately 44°N because this affects the direction of the shadow!)

- Determine the slant height of the tower.
- How high is the tip of the tower above the ground?

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### 3 Answers

Let A be the top of the tower, B the base of the tower and C the tip of the shadow.

Since the tower is in the northern hemisphere, the sun will be to the south and the shadow will be towards the north. Since the tower leans at an angle of `5.5^@` towards the south, the angle at the base is `95.5^@` as measured from the north.

Then `m/_B=95.5^@,m/_C=32^@,a=84.02m`

(1) We are asked to determine the length c in our model:

` `Since the angle sum of a triangle is `180^@` we have `m/_A=52.5^@`

Then we apply the Law of Sines: `84.02/(sin52.5^@)=c/sin32^@`

`c=(84.02sin32^@)/sin52.5^@~~56.12`

**Thus the slant height of the tower is approximately 56.12m**

(2) We are asked to determine the height of the tip of the tower to the ground:

Dropping an altitude from A to the ground at D we have d=c=56.12 and `m/_ABD=84.5^@` Let AD=b

`sin84.5=b/56.12 ==>b=56.12sin84.5~~55.86`

**So the height of the tower above the ground is approximately 55.86m**

Calculate the height of a lean-to roof that has a slanted height of 325 cm and an angle of elevation of 22°. Round your answer to the nearest cm.

Let AB be tower , A top and B is base of the tower. Tower lean towards south,that AB inclined to horizontal at 5.5 degree, south direction. Drop a perpendicular form A , meet to ground at D.So angle ABD=90-5.5=84.5

ABD is right angle triangle , angle D is right angle.AD is actual hight of the tower above ground.

When angle of elevation of the sun is 32 degree, shadow meet ground at C (say). length of the shadow is CD=84.02 m

In right angle triangle ADC,

tan(32)=AD/CD

AD= CD tan(32)

AD=84.02 x .625=52.502 m

In right angle triangle ADB,

Sin(90-5.5)=AD/AB

AB=AD/Sin(84.5)

AB=52.502 / .995 = =52.745m

Thus tip of the tower above the ground=52.502m

Slant hight of the tower=52.745m