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Chapter 10: Problem 27

Two vertical towers have heights \(9 \mathrm{~m}\) and \(17.2 \mathrm{~m}\) and are \(42 \mathrm{~m}\) apart. (a) Calculate the angle of elevation from the base of the shorter tower to the top of the taller tower. (b) Calculate the angle of depression from the top of the taller tower to the top of the shorter tower.

Short Answer

Expert verified
Question: Calculate the angle of elevation from the base of the shorter tower to the top of the taller tower and the angle of depression from the top of the taller tower to the top of the shorter tower, given that the height of the shorter tower is 9m, the height of the taller tower is 17.2m, and the distance between the towers is 42m. Answer: The angle of elevation from the base of the shorter tower to the top of the taller tower is approximately 11.04°, and the angle of depression from the top of the taller tower to the top of the shorter tower is approximately 12.16°.

Step by step solution

01

Draw the diagram

First, we will draw the diagram representing the given scenario. From the base of the shorter tower, draw a horizontal line segmentABCDE, making an angle with the vertical shorter tower. Draw the vertical taller tower parallel to the shorter tower, with a distance of 42m between them. Label the base of the shorter tower as A, top of the shorter tower as B, the base of the taller tower as C, the top of the taller tower as D, and the intersection of the horizontal line segment and the taller tower as E.
02

Identify the given information and the unknowns

The given information is as follows: - Height of the shorter tower, AB = 9m - Height of the taller tower, CD = 17.2m - Distance between the towers, AC = 42m The unknowns are: - Angle of Elevation from A to D, ∠BAD - Angle of Depression from D to B, ∠DBE
03

Calculate height difference and find triangles

In order to find the angle of elevation and depression, we first need to calculate the height difference between the two towers. Height difference, DE = CD - AB = 17.2 - 9 = 8.2m Now, we can identify two right-angled triangles: 1. Right Triangle ΔABE with AB as one leg, BE as the other leg, and AE as the hypotenuse. 2. Right Triangle ΔACD with AD as one leg, AC as the other leg, and AE as the hypotenuse.
04

Calculate the angle of elevation

In right triangle ΔACD, we will use the tangent trigonometric ratio to determine the angle of elevation ∠BAD. tan(∠BAD) = \frac{opposite \ side}{adjacent \ side} = \frac{AD}{AC} So, tan(∠BAD) = \frac{8.2}{42} Now, calculate the angle ∠BAD: ∠BAD = arctan(\frac{8.2}{42}) Therefore, the angle of elevation is ∠BAD ≈ 11.04°.
05

Calculate the angle of depression

In right triangle ΔABE, we will use the tangent trigonometric ratio to determine the angle of depression ∠DBE. tan(∠DBE) = \frac{opposite \ side}{adjacent \ side} = \frac{BE}{AC} But, BE = AB, So, tan(∠DBE) = \frac{9}{42} Now, calculate the angle ∠DBE: ∠DBE = arctan(\frac{9}{42}) Therefore, the angle of depression is ∠DBE ≈ 12.16°. The final answers are: (a) The angle of elevation from the base of the shorter tower to the top of the taller tower is approximately 11.04°. (b) The angle of depression from the top of the taller tower to the top of the shorter tower is approximately 12.16°.

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