The teet of two vertical poles of height 3m and 7m are in line with a point P on the ground. The smaller pole being between the taller pole and P and at distance of 20m from P the angle of elevation of the top T of the taller pole from the top R of the smaller pole is 30°.find the distance RT distance of the foot of the taller pole P correct to 3 significant figure angles of elevation of T from P to the nearest degree.

In geometry problems, we need to start drawing a sketch according to instructions.

On the sketch, label the unknowns, and hence deduce the equations/formulas that may help to solve the problem.
Start by sketching according to given information.

Geometry will always be difficult if one does not have the habit or skill of drawing a sketch to display given information.

To draw the sketch, analyse sentences one at a time. I'll get you started:
"The teet of two vertical poles of height 3m and 7m are in line with a point P on the ground. "

Draw three points along a horizontal line. Name the left most point P.

"The smaller pole being between the taller pole and P and at distance of 20m from P"

From the middle point, draw a vertical line (upwards) for the 3m pole, and from the rightmost point, draw a vertical line for the 7m pole. Mark 20m between point P and the smaller (3m) pole.

" the angle of elevation of the top T of the taller pole from the top R of the smaller pole is 30°"

Name the top of the 7m pole T, and the top of the 3m pole R.
The angle of elevation from R to T is 30°. Mark that on your diagram.

Now for the solution part, draw a horizontal line through R to intersect the 7m pole at S.

Extract the right triangle RST and indicate the known measurements:
angle of elevation = 30 degrees
height ST=7-3=4m
angle RST=90 degrees.

Hence we have a right triangle of which two of the dimensions are known. Solve for the distance (adjacent side to 30°) RS.

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To solve this problem, we can draw a diagram and use trigonometry. Let's label the points on the diagram for easier explanation:

- Let T be the top of the taller pole.
- Let R be the top of the smaller pole.
- Let P be the point on the ground where the poles and the observer are aligned.

Now, let's determine the distance RT using the information provided.

1. Draw a triangle TRP with the poles as two of its sides and the distance between P and the smaller pole as the third side.

2. We know that the height of the taller pole is 7m, and the height of the smaller pole is 3m. Since the angle of elevation from R to T is 30°, we can use trigonometry to find the length of RT.

3. In the triangle TRP, we have a right triangle TRT', where T' is the point on RT directly below point R.

4. The tangent of angle T'RP is equal to the opposite side (PR) divided by the adjacent side (RT'). Therefore, tan(T'RP) = 3 / RT'.

5. Solving for RT', we get RT' = 3 / tan(T'RP).

6. Since angle T'RP is equal to 90° - 30° = 60° (using complementary angles), we can now calculate RT'.

7. Plug in the values to find RT': RT' = 3 / tan(60°) = 3 / √3 ≈ 1.732m.

So, the distance RT is approximately 1.732m.

Next, let's find the distance of the foot of the taller pole, P, correct to 3 significant figures.

8. We know that the distance between P and the smaller pole is 20m, and we can use this information to find the distance of the foot of the taller pole, P.

9. Considering the triangle TRP, we have a right triangle TPT", where T" is the point on PT directly below point T.

10. The tangent of angle T"PT is equal to the opposite side (PP') divided by the adjacent side (PT"). Therefore, tan(T"PT) = 7 / PT".

11. Solving for PT", we can use the tangent function to find PT": PT" = 7 / tan(T"PT).

12. Since angle T"PT is equal to 90° - 30° = 60° (using complementary angles), we can calculate PT".

13. Plug in the values to find PT": PT" = 7 / tan(60°) = 7 / √3 ≈ 4.041m.

Thus, the distance of the foot of the taller pole, P, is approximately 4.041m, correct to 3 significant figures.

Finally, let's find the angles of elevation of T from P to the nearest degree.

14. Since we know the distances PT and RT, we can determine the angles of elevation using trigonometry.

15. The angle of elevation TPQ (from P) is equal to the inverse tangent of the height (7 m) divided by the distance PT.

16. Using the inverse tangent function, we can calculate TPQ: TPQ = arctan(7/PT).

17. Plugging in the value for PT (approximately 4.041m) into the equation, we get TPQ = arctan(7/4.041) ≈ 59°.

Therefore, the angle of elevation of T from P is approximately 59° to the nearest degree.