Height And Distance

Step 1: Let tower height be . Distance to base from A = .
Step 2: Distance to base from B = .
Step 3: Distance AB = . Time for AB = 20s.
Step 4: Speed . Time for B to base = .
Step 5: Rationalize: .

Step 1: Since the angle of elevation of the top of the pole ( ) is the same from all vertices, the foot of the pole must be the circumcenter ( ) of .
Step 2: In (where is the top), and .
Step 3: .

Step 1: Understanding the Concept:
In problems involving spheres subtending angles at an eye, the angle subtended ( ) is the angle between the two tangent lines from the eye to the sphere.
The observer's eye, the center of the sphere, and the point of tangency form a right-angled triangle.
The total height of the topmost point is the height of the center of the sphere plus the radius of the sphere.
Step 2: Key Formula or Approach:
1. If distance from eye to center is and radius is , then , where is the subtended angle.
2. Height of center , where is the angle of elevation.
3. Total Height .
Step 3: Detailed Explanation:
Given radius .
Subtended angle .
From the geometry: .
Angle of elevation of the center .
Height of the center of the balloon ( ):

Using the sine sum formula :

Multiply numerator and denominator by :

Substituting back into the height formula:

Height of the topmost point ( ):

Factor out 8:

Step 4: Final Answer:
The height of the top most point is meters.

Step 1: Speed . Distance .
Step 2: Let height be . In the first position, .
Step 3: In the second position, .
Step 4: .
Step 5: .

Step 1: Midpoint distance m. Heights are and .
Step 2: and .
Step 3: .
Step 4: .
Step 5: .