Angle of elevation

The angle of elevation of the top of an unfinished pillar at a point 150 m from its base is 30°. If the angle of elevation at the same point is to be 45°, then the pillar has to be raised to a height of how many meters?

Correct answer:

y =  63.4 m

Step-by-step explanation:

x=150 m α=30 ∘ β=45 ∘  tan α = h : x  h=x⋅ tanα=x⋅ tan30° =150⋅ tan30° =150⋅ 0.57735=86.60254 m  tan β = (h+y) : x  x⋅ tan β = h+y  y=x⋅ tanβ−h=x⋅ tan45° −h=150⋅ tan45° −86.6025=150⋅ 1−86.6025=63.4=63.4 m

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Dr. Math
To determine how much taller the unfinished pillar needs to be so that the angle of elevation from 150 meters away changes from 30° to 45°, follow these steps:

1. Initial Setup:
  - Let h be the current height of the pillar.
  - The distance from the observation point to the base of the pillar is 150 meters.
  - The initial angle of elevation is 30°.

2. Calculate the Current Height ( h ):

  Using the tangent function for the angle of elevation:
 
  tan(30°) = h/150
 

 
  h = 150 × tan(30°)
 

 
  h = 150 × 1/ √3 ≈ 150 × 0.577 ≈ 86.6 meters
 


3. Determine the Required Height ( H ) for 45° Elevation:

  For the angle of elevation to be 45°:
 
  tan(45°) = H/150
 

 
  H = 150 × tan(45°) = 150 × 1 = 150 meters
 


4. Calculate the Additional Height Needed:

  Subtract the current height from the required height:
 
  Additional Height = H - h = 150 - 86.6 = 63.4 meters
 


However, let's verify the calculations more precisely:

- Precise Calculation of h :
 
 h = 150 × tan(30°) = 150 × 1/ √3 = 150/ √3 = 50 √3 ≈ 86.6 meters
 


- Precise Calculation of Additional Height:
 
 Additional Height = 150 - 50 √3 ≈ 150 - 86.6 = 63.4 meters
 


But let's express the additional height exactly:

Additional Height = 150 - 50 √3 = 50(3 - √3 ) meters


However, based on the initial approximation, the additional height is approximately 63.4 meters.

But reviewing the exact calculation:

tan(30°) = 1/ √3 ⇒ h = 150 × 1/ √3 = 50 √3 meters


tan(45°) = 1 ⇒ H = 150 × 1 = 150 meters


Additional Height = H - h = 150 - 50 √3 meters


Numerically:

50 √3 ≈ 86.6 meters


150 - 86.6 = 63.4 meters


Therefore, the pillar needs to be raised by approximately 63.4 meters.

However, let's consider the exact form for precision:

Additional Height = 150 - 50 √3 = 50(3 - √3 ) meters


But since the problem expects a numerical answer, we'll use the approximate value.

Final Answer:

63.4 meters







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