The Leaning tower of Pisa leans toward the south at an angle of about 5.5°. one day, it;s shadow is 90m long, and the anlge of elevation from the tip of the shadow to the top of the tower is 32°. what is the slant height of the tower?
nice try, but I'm not falling for it.
which direction is the sun shining from?
If directly from the north (unlikely) then the law of sines says that the slant height s can be found via
s/sin32° = 90/sin52.5°
To find the slant height of the tower, we can use trigonometry.
Let's label the given information:
Angle of inclination of the tower = 5.5° (angle between the vertical and the tower)
Length of the shadow = 90m
Angle of elevation = 32° (angle between the ground and the line from the tip of the shadow to the top of the tower)
To find the slant height, we need to calculate the height of the tower first.
Step 1: Find the height of the tower using the given information.
We can form a right-angled triangle with the height of the tower as the vertical side, the shadow as the horizontal side, and the slant height as the hypotenuse.
In the triangle, the angle between the horizontal side (shadow) and the hypotenuse is (90° - 32°) since the sum of angles in a triangle is 180°.
Step 2: Calculate the height of the tower.
Using trigonometry, we can use the formula:
height = shadow * tan(angle between the horizontal side and the hypotenuse)
height = 90m * tan(90° - 32°)
height = 90m * tan(58°)
Calculating the value using a calculator gives us:
height ≈ 90m * 1.5847
Therefore, the height of the tower is approximately 142.63m.
Step 3: Calculate the slant height of the tower.
In the same right-angled triangle, the angle between the vertical side (height) and the hypotenuse (slant height) is the angle of inclination of the tower (5.5°).
Using trigonometry, we can use the formula:
slant height = height / cos(angle of inclination of the tower)
slant height = 142.63m / cos(5.5°)
Calculating the value using a calculator gives us:
slant height ≈ 142.63m / 0.9956
Therefore, the slant height of the tower is approximately 143.30m.
To find the slant height of the tower, we can use trigonometry. Let's break down the problem step by step.
Step 1: Understand the problem.
The problem states that the Leaning Tower of Pisa leans toward the south at an angle of about 5.5°. We are given the length of the shadow cast by the tower (90m) and the angle of elevation from the tip of the shadow to the top of the tower (32°). We need to calculate the slant height of the tower.
Step 2: Visualize the problem.
Imagine a right-angled triangle formed by the shadow of the tower, the height of the tower, and the slant height of the tower. The angle of elevation from the tip of the shadow to the top of the tower gives us one angle in the triangle, and the angle at the base where the shadow meets the ground is the complement of the angle of the tower's lean (90° - 5.5°).
Step 3: Identify the relevant trigonometric functions.
We can use the tangent function to relate the angle of elevation and the height of the tower, and the sine function to relate the angle of the tower's lean and the slant height of the tower.
Step 4: Calculate the height of the tower.
Using the tangent function, we can determine the height of the tower by using the angle of elevation and the length of the shadow:
tan(32°) = height of tower / length of shadow
height of tower = length of shadow * tan(32°)
height of tower = 90m * tan(32°)
Step 5: Calculate the slant height of the tower.
Using the sine function, we can determine the slant height of the tower by using the angle of the tower's lean and the height of the tower:
sin(5.5°) = slant height of tower / height of tower
slant height of tower = height of tower * sin(5.5°)
slant height of tower = (90m * tan(32°)) * sin(5.5°)
Step 6: Calculate the final answer.
To find the slant height of the tower, plug in the values we already know:
slant height of tower = (90m * tan(32°)) * sin(5.5°)
slant height of tower ≈ 90m * 0.624 * 0.096
slant height of tower ≈ 5.35m
Therefore, the slant height of the Leaning Tower of Pisa is approximately 5.35 meters.