On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower. Can any one and psl?

Question

Answer

Answer

Answer

Here's your answer bud,
Given:
Angles are complementary therefore one will be 'theta' and the other will be '90-theta'

Tan theta = AB/16 where AB is the height of tower------1
Tan (90-theta)= AB/4------2
Tan (90-theta)= cot theta

Multiply 1 and 2

Tan theta * cot theta = AB/ 16 * AB/4

1 = AB^2/64

64= AB^2

AB=8
Upvote and lemme know if that helped :)

Answer

Problem:
On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

Solution:
To solve this problem, we can use trigonometric ratios and the concept of complementary angles. Let's break down the solution into steps:

Step 1: Identifying the given information
- Two points, C and D, are on a straight line passing through the foot of the tower.
- The distance from the foot of the tower to point C is 4 m.
- The distance from the foot of the tower to point D is 16 m.
- The angles of elevation from points C and D to the top of the tower are complementary.

Step 2: Drawing a diagram
To better understand the problem, let's draw a diagram.

```
C T D
\ | /
\ |h /
\ | /
\ | /
\ | /
\ | /
\ | /
\|/
F
```

In the diagram, T represents the top of the tower, F represents the foot of the tower, and h represents the height of the tower.

Step 3: Applying trigonometry
We can use trigonometric ratios to find the height of the tower.

From the diagram, we can see that tan(angle of elevation from C) = h/4 and tan(angle of elevation from D) = h/16.

Since the angles of elevation from C and D are complementary, we have the equation:

tan(angle of elevation from C) = cot(angle of elevation from D)

Using the definition of cotangent, we can rewrite the equation as:

h/4 = 1/tan(angle of elevation from D)

Step 4: Evaluating the trigonometric expression
To evaluate the trigonometric expression, we need to find the value of tan(angle of elevation from D).

From trigonometry, we know that tan(angle of elevation) = opposite/adjacent. In this case, the opposite side is the height of the tower (h) and the adjacent side is the distance from the foot of the tower to point D (16 m).

Therefore, tan(angle of elevation from D) = h/16

Step 5: Solving the equation
Now we can substitute the value of tan(angle of elevation from D) into the equation and solve for h:

h/4 = 1/(h/16)

Multiplying both sides by 4 and 16, we get:

16h = 4*16

16h = 64

Dividing both sides by 16, we find:

h = 4

Step 6: Final answer
The height of the tower is 4 meters.

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