Ainsley’sandCaleb’sapartmentbuildingsareexactlythesameheight. Ainsley measures the distance between the buildings as 51 m and observes that the angle of depression from the roof of her building to the bottom of Caleb’s is about 64°. How tall, to the nearest metre, is each building?
review your basic trig functions. You will see that the height h can be found using
h/51 = tan 64°
To find the height of each building, we can use trigonometry and the concept of the angle of depression.
First, let's define the given information:
- Ainsley's distance measurement between the buildings = 51 m
- The angle of depression from Ainsley's roof to the bottom of Caleb's building = 64°
Now, let's draw a diagram to visualize the problem. Imagine two vertical lines representing the buildings, with Ainsley's building on the left and Caleb's building on the right. We also have an angle of 64°, formed by Ainsley's line of sight from her roof to the bottom of Caleb's building.
| Ainsley | Caleb |
|-----T----|----|
\ | /
\ | /
\ |h1 / <- Distance = 51 m
\ | /
\ | /
\ |/
\ |θ
\|
In this diagram:
- T represents the top of Ainsley's building
- h1 represents the height of Ainsley's building
- θ represents the angle of depression (64°)
Using trigonometry, we know that the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the height of Ainsley's building is the opposite side, and the distance between the buildings is the adjacent side.
Thus, we can use the tangent function to determine h1:
tan(θ) = h1 / Distance
Rearranging the equation:
h1 = Distance * tan(θ)
Plugging in the values:
h1 = 51 m * tan(64°)
h1 ≈ 115.76 m (rounded to 2 decimal places)
Therefore, the height of Ainsley's building is approximately 115.76 meters.
Since both buildings are the same height, Caleb's building would also be approximately 115.76 meters tall.