Two points, A and D, are in a horizontal line with the foot of a tower, CB, and on opposite sides. The distance between A and D is 535.4 feet while the angles of elevation of the top B are 12 degrees 46 minutes from A and 18 degrees 38 minutes from D. Let the perpendicular through D to line AB produced meet it at E and find the length of CB
To find the length of CB, we need to use trigonometry. Let's break down the given information and steps to solve for CB:
1. We have two points, A and D, in a horizontal line with the foot of the tower CB.
2. The distance between A and D is 535.4 feet.
3. The angle of elevation of point B from point A is 12 degrees 46 minutes.
4. The angle of elevation of point B from point D is 18 degrees 38 minutes.
Now, let's identify the triangle we can form and use trigonometric ratios to solve for CB:
1. Draw a diagram with points A, D, E, and B. AB is the horizontal line where A and D lie, and ED is the perpendicular line.
2. We can form a right triangle ABE, where AB is the base, AE is the height, and angle ABE is a right angle.
3. In triangle ABE, we can find the length of AE using trigonometry.
4. From point A, the angle of elevation to B is 12 degrees 46 minutes. This means the angle ABE in triangle ABE is 90 degrees minus 12 degrees 46 minutes.
5. Use the sine function to find the length of AE: sin(ABE) = AE/AB. Rearrange the formula to AE = AB * sin(ABE).
6. Similarly, find the length of DE using trigonometry. From point D, the angle of elevation to B is 18 degrees 38 minutes. This means the angle DBE in triangle DBE is 90 degrees minus 18 degrees 38 minutes.
7. Use the sine function to find the length of DE: sin(DBE) = DE/DB. Rearrange the formula to DE = DB * sin(DBE).
8. The distance between A and D is given as 535.4 feet. Therefore, AD = AE + DE.
9. Now, we have the values of AE, DE, and AD. To find CB, we need to find the length of CD.
10. CD is the perpendicular from D to line AB produced. In triangle ACD, we can use the Pythagorean theorem to find CD.
11. Apply the Pythagorean theorem: AD^2 = AC^2 + CD^2.
12. Rearrange the formula to CD = √(AD^2 - AC^2).
13. Plug in the values of AD and AE obtained earlier to find CD.
14. Finally, subtract CD from AD to find CB: CB = AD - CD.
By following these steps and computing the calculations, you can find the length of CB.