5. The Leaning Tower of Pisa leans toward the south at an angle of 5.5 degrees. One day near noon its shadow was measured to be 84.02m long and the angle of elevation from the tip of the shadow to the following, assume that the Tower is like a pole stuck in the ground, that is, it has negligible width.

a. Determine the slant of the tower

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

To determine the slant of the tower, we can use trigonometry. Let's denote the height of the tower as "h" and the slant distance from the tip of the tower to the shadow as "s."

a. To find the slant distance, we can use the tangent of the angle of elevation. The tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the tower (h), and the adjacent side is the shadow length (84.02m). Therefore, we can write:

tangent(5.5 degrees) = h / 84.02m

Now we can solve for h:

h = tangent(5.5 degrees) * 84.02m

Using a scientific calculator, we can find the tangent of 5.5 degrees, which is approximately 0.096179. Plugging this value into the equation, we have:

h ≈ 0.096179 * 84.02m

Calculating this, we get:

h ≈ 8.16m

Therefore, the height of the tower (slant distance) is approximately 8.16m.

b. Next, we can determine the actual height of the tip of the tower above the ground. To do this, we need to consider the vertical height, which is perpendicular to the ground.

We can use the sine of the angle of elevation to find the vertical height. The sine of an angle is the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height of the tower (h), and the hypotenuse is the slant distance (s). Therefore, we can write:

sine(5.5 degrees) = h / s

Now substituting the calculated values:

sine(5.5 degrees) = 8.16m / s

Using a scientific calculator, we can find the sine of 5.5 degrees, which is approximately 0.096640. Plugging this value into the equation, we have:

0.096640 = 8.16m / s

Solving for s:

s = 8.16m / 0.096640

Calculating this, we get:

s ≈ 84.49m

Therefore, the tip of the tower is approximately 84.49 meters above the ground.