Question

To estimate the height of a house Katie stood a certain distance from the house and determined that the angle of elevation to the top of the house was 32 degrees. Katie then moved 200 feet closer to the house along a level street and determined the angle of elevation was 42 degrees. What is the height of the house.Step by Step Explanation .. Please

Answer 1

the height of the house is $408ft$ .

Step-by-step explanation:

Here we have , To estimate the height of a house Katie stood a certain distance from the house and determined that the angle of elevation to the top of the house was 32 degrees. Katie then moved 200 feet closer to the house along a level street and determined the angle of elevation was 42 degrees. We need to find  What is the height of the house . Let's find out:

Let  y is the unknown height of the house, and x is the unknown number of feet she is standing from the house.

Distance of house from point A( initial point ) = x ft

Distance of house from point B( when she traveled 200 ft towards street  = x-200 ft

Now , According to question these scenarios are of right angle triangle as

At point A

⇒ $Tan32 =\frac{Perpendicular}{Base}= \frac{y}{x}$

⇒ $Tan32 = \frac{y}{x}$

⇒ $y=x(Tan32 )$       ..................(1)

Also , At point B

⇒ $Tan42 = \frac{y}{x-200}$

⇒ $y=(x-200)(Tan42)$     ..............(2)

Equating both equations:

⇒ $(x-200)(Tan42) = x(Tan32)$

⇒ $x(Tan42-Tan32)=Tan42(200)$

⇒ $x=\frac{Tan42(200)}{(Tan42-Tan32)}$

⇒ $x=653ft$

Putting  $x=653ft$ in  $y=x(Tan32 )$  we get:

⇒  $y=x(Tan32 )$    

⇒  $y=653(Tan32 )$

⇒  $y=408ft$

Therefore , the height of the house is $408ft$ .