10. A person notices that the angle of elevation from where he is standing to the top of a pole is 40°. Then he backs up
12 feet and notices that the angle of elevation to the top of the pole is now 32°. How tall is the pole?
draw the diagram. It should be clear that if the height is h,
h cot32° - h cot40° = 12
To find the height of the pole, we can use trigonometry and the concept of similar triangles. Here's how you can solve this problem step by step:
Step 1: Draw a diagram: Start by drawing a diagram to represent the situation. Draw a straight line to represent the ground and a perpendicular line to represent the pole. Label the angle of elevation (40° and 32°) and the distance between the person and the pole (12 feet).
Step 2: Identify similar triangles: In the diagram, we have two right triangles. The first triangle consists of the person, the pole, and the point where the person is standing. The second triangle consists of the person, the pole, and the point where the person is standing after moving back.
Step 3: Set up proportions: Since the triangles are similar, we can set up a proportion to find the height of the pole. Let h be the height of the pole. In the first triangle, the opposite side is h, and the adjacent side is the distance between the person and the pole. In the second triangle, the opposite side remains h, and the adjacent side is the distance between the person (backed up) and the pole.
Using the trigonometric ratio tangent (opposite/adjacent), we can write the following equations:
tan(40°) = h / 12 (first triangle)
tan(32°) = h / (12 + 12) (second triangle)
Step 4: Solve the equations: Now we can solve these two equations simultaneously to find the height of the pole (h).
From the first equation, we can isolate h:
h = tan(40°) * 12
From the second equation, we can isolate h:
h = tan(32°) * 24
Step 5: Calculate the height of the pole: Substitute the values of tangent for each angle and calculate the height.
h = 0.8391 * 12 ≈ 10.07 feet (from the first equation)
h = 0.6249 * 24 ≈ 14.998 feet (from the second equation)
Since the height of the pole should be the same in both cases, we can take the average of these two values:
Height ≈ (10.07 + 14.998) / 2 ≈ 12.534 feet
Therefore, the height of the pole is approximately 12.534 feet.