From the roof of Fiona's house, the angle of elevation of the top of Eric's house is 15 degrees, and the angle of depression to the bottom of Eric's house is 20 degrees. The houses... From the roof of Fiona's house, the angle of elevation of the top of Eric's house is 15 degrees, and the angle of depression to the bottom of Eric's house is 20 degrees. The houses are 10 meters apart. Calculate the height of Eric's house. Read more

Steps to Solve

1. Identify the Variables

Let:

• ( d = 10 ) m (distance between the two houses)
• ( h ) = height of Eric's house
• ( h_f ) = height of Fiona's house

2. Draw the Right Triangles

Two right triangles can be formed:

• Triangle for the angle of elevation (15°) to the top of Eric's house.
• Triangle for the angle of depression (20°) to the bottom of Eric's house.

3. Establish the Height from Fiona's House

Using trigonometry for the top of Eric's house:

• The opposite side is the difference in height ( h - h_f )
• The adjacent side is the distance ( d ):

Using the tangent function:

$$ \tan(15^\circ) = \frac{h - h_f}{10} $$

Therefore,

$$ h - h_f = 10 \tan(15^\circ) $$

4. Establish the Height from the Bottom of Eric's House

Using trigonometry for the bottom of Eric's house:

• The opposite side is ( h_f )

Using the tangent function:

$$ \tan(20^\circ) = \frac{h_f}{10} $$

Hence,

$$ h_f = 10 \tan(20^\circ) $$

5. Set Up the Equation

Substituting ( h_f ) into the equation for ( h ):

$$ h - 10 \tan(20^\circ) = 10 \tan(15^\circ) $$

Therefore,

$$ h = 10 \tan(15^\circ) + 10 \tan(20^\circ) $$

6. Calculate the Values

Calculate the values of (\tan(15^\circ)) and (\tan(20^\circ)) using a calculator:

$$ \tan(15^\circ) \approx 0.2679 $$ $$ \tan(20^\circ) \approx 0.3640 $$

Now substitute these values:

$$ h = 10(0.2679) + 10(0.3640) $$

Calculate:

$$ h \approx 2.679 + 3.640 = 6.319 \text{ m} $$