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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.
The tower makes an angle of 78.5 degrees with the ground.
Let x be the length of the shadow.
Side A: 84.02m
Side B: 8.1m
Side C: 82.9m
Angle A: 5.5 degrees
Angle B: 78.5 degrees
Angle C: 96 degrees
b. How high is the tip of the tower above the ground?
The tip of the tower is 82.9 meters above the ground.
To determine the slant of the tower, we will use the trigonometric relationship between the angles and sides of a right triangle.
In this case, we have angle A as the angle of elevation, side A as the height of the tower, and side B as the length of the shadow. Using the tangent function, we can calculate the slant of the tower.
tan(A) = opposite/adjacent
tan(78.5) = Side A/Side B
tan(78.5) = x/84.02
x = 84.02 * tan(78.5)
x = 82.9 meters
Therefore, the slant of the tower is approximately 82.9 meters.
To determine the height of the tip of the tower above the ground, we can use the sine function with angle A and side C.
sin(A) = opposite/hypotenuse
sin(5.5) = Side A/Side C
Side A = 82.9 * sin(5.5)
Side A = 8.1 meters
Therefore, the tip of the tower is approximately 82.9 meters above the ground.
a. To determine the slant of the tower, we can use trigonometry. Since we are given the length of the shadow (side A), we can use the tangent function.
tan(angle A) = opposite/adjacent
tan(5.5 degrees) = opposite/84.02m
Using a calculator, we can find that the opposite side (height of the tower) is approximately 8.1m.
Therefore, the slant of the tower is 8.1m.
b. To determine how high the tip of the tower is above the ground, we can use trigonometry again. This time, we will use the sine function since we know the opposite side (side A) and the angle opposite to it (angle C).
sin(angle C) = opposite/hypotenuse
sin(96 degrees) = 8.1m/hypotenuse
Solving for the hypotenuse (height of the tower), we find:
hypotenuse = (8.1m) / sin(96 degrees)
hypotenuse ≈ 82.9m
Therefore, the tip of the tower is approximately 82.9 meters above the ground.