The angle of depression from the top of a lighthouse across the street is 65 degrees. the angle of depression from the top of the lighthouse to the top of a house is 28 degrees. the distance from the lighthouse to the house is 37 feet. what is the height of the house?
The angle of depression from the top of a lighthouse across the street is 65 degrees.
the angle from the lighthouse to what? And across the street from what?
All we know is that the lighthouse is
37 tan28° = 19.67 ft taller than the house.
To find the height of the house, we can use the trigonometric concept of the angle of depression and the distance between the lighthouse and the house.
Let's break down the problem. We have two angles of depression: one from the top of the lighthouse across the street, and the other from the top of the lighthouse to the top of the house.
The angle of depression from the lighthouse to the top of the house is 28 degrees, which means that if we draw a horizontal line from the top of the lighthouse, the angle between this line and the line connecting the lighthouse and the top of the house will be 28 degrees.
The angle of depression from the top of the lighthouse across the street is 65 degrees. Again, if we draw a horizontal line from the top of the lighthouse, the angle between this line and the line connecting the lighthouse and the street will be 65 degrees.
Now, let's represent the problem:
Lighthouse
/ \
/ \
/ h \
H | _______________ | S
|___________________|
In the above figure:
Lighthouse = L
House = H
Street = S
Angle of depression from L to H = 28 degrees
Angle of depression from L to S = 65 degrees
Distance from L to H = 37 feet
Now, let's apply some trigonometry to find the height of the house, H.
Step 1: Determine the horizontal distance from the lighthouse to the house.
We can use the tangent function to find this distance.
tan(65 degrees) = H/S
H = S * tan(65 degrees)
H = 37 feet * tan(65 degrees)
H ≈ 37 feet * 2.1445 (rounded to 4 decimal places)
H ≈ 79.29 feet (rounded to 2 decimal places)
Step 2: Find the vertical distance from the lighthouse to the top of the house.
Again, we will use the tangent function.
tan(28 degrees) = h/S
h = S * tan(28 degrees)
h = 37 feet * tan(28 degrees)
h ≈ 37 feet * 0.5317 (rounded to 4 decimal places)
h ≈ 19.67 feet (rounded to 2 decimal places)
Step 3: Calculate the height of the house.
The height of the house is the sum of the vertical distance from the lighthouse to the top of the house (h) and the height of the lighthouse (H).
Height of the house = h + H
Height of the house ≈ 19.67 feet + 79.29 feet
Height of the house ≈ 98.96 feet (rounded to 2 decimal places)
Therefore, the height of the house is approximately 98.96 feet.