From the window of one building, Matt finds the angle of elevation of the top of a second building is 33° and the angle of depression to the bottom of the same building is 55°. The buildings are 25 meters apart. Find the height of the second building to the nearest meter.
To find the height of the second building, we can use trigonometry. Let's call the height of the second building "h".
First, let's consider the angle of elevation. The angle of elevation is the angle between the horizontal line and the line of sight from the observer to the top of the second building. We can use the tangent function to find the height of the second building:
tan(33°) = h / 25
To solve for "h", we can multiply both sides of the equation by 25:
25 * tan(33°) = h
h ≈ 14.6 meters (rounded to the nearest meter)
Now let's consider the angle of depression. The angle of depression is the angle between the horizontal line and the line of sight from the observer to the bottom of the second building. Since the observer is on the first building, they are looking downward. We can use the tangent function again to find the height of the second building:
tan(55°) = h / 25
To solve for "h", we can multiply both sides of the equation by 25:
25 * tan(55°) = h
h ≈ 35.1 meters (rounded to the nearest meter)
Since the height of the second building cannot be negative, the height is approximately 35.1 meters (rounded to the nearest meter).
To find the height of the second building, we can use the concept of trigonometry.
Let's label the height of the second building as 'h'.
First, let's analyze the situation. We know that Matt is looking at the top of the second building, so we have the angle of elevation, which is 33°. This angle is formed between the horizontal line of sight from Matt's eye to the top of the building and the line of sight from Matt's eye to the top of the building.
We also know that Matt is looking at the bottom of the second building, so we have the angle of depression, which is 55°. This angle is formed between the horizontal line of sight from Matt's eye to the bottom of the building and the line of sight from Matt's eye to the bottom of the building.
Now, let's draw a diagram:
*
/|
/ |
/ |h
/ |
/___|___
/ θ|
/_____|
In the diagram, '*' represents Matt's position in the first building. The second building is represented by a vertical line segment labeled 'h'. The angle of elevation is labeled 'θ' (33°), and the angle of depression is labeled 'θ' (55°).
Since the buildings are 25 meters apart, we have a right triangle formed by the horizontal line connecting Matt's position to the bottom of the second building, the vertical line representing the height of the second building, and the hypotenuse connecting Matt's position to the top of the second building.
Now we can use the trigonometric function "tangent" to solve for the height of the second building.
1. Tangent of the angle of elevation:
tan(33°) = h / 25
Rearranging the equation, we get:
h = 25 * tan(33°)
2. Tangent of the angle of depression:
tan(55°) = h / 25
Rearranging the equation, we get:
h = 25 * tan(55°)
Using a scientific calculator, calculate the value of 'h' using the equations above.
Finally, round the value of 'h' to the nearest whole number to find the height of the second building.
the height is just
25(tan33°+tan55°)