The angle of elevation to the top of a building from a point on the ground is 39 degrees. From a point 50 feet closer to the building, the angle of elevation is 48 degrees. What is the height of the building?
draw the figure. You have two triangles. let x be the distance to the closer point, from the building.
Tan48=h/x
Tan39=h/(x+50)
or
x=h/tan48 and
x=h/tan39 -50
set them equal, and solve for h.
To solve this problem, we can use trigonometry, specifically the tangent function.
Let's assume that the distance between the initial point on the ground and the base of the building is 'x' feet. This would mean that the distance between the second point (50 feet closer to the building) and the base of the building is 'x - 50' feet.
Now, let's consider the right triangles formed. The angle of elevation to the top of the building from the initial point is 39 degrees, and the angle of elevation from the second point is 48 degrees. The opposite side of the right triangle is the height of the building.
Using the tangent function, we can write the following relationships:
For the initial point:
tan(39 degrees) = height of the building / x
For the second point:
tan(48 degrees) = height of the building / (x - 50)
Now, we can solve for the height of the building by setting up these two equations as a system of equations and solving them simultaneously.
First, rearrange the equations:
height of the building = x * tan(39 degrees) ...(1)
height of the building = (x - 50) * tan(48 degrees) ...(2)
Since both expressions are equal to the height of the building, we can set them equal to each other:
x * tan(39 degrees) = (x - 50) * tan(48 degrees)
Now, we can solve for 'x' and then substitute it back into equation (1) to calculate the height of the building.
Let's do the calculations:
tan(39 degrees) = 0.809 (rounded to three decimal places)
tan(48 degrees) = 1.110 (rounded to three decimal places)
Divide both sides of the equation by tan(48 degrees):
x = (x - 50) * 1.110 / 0.809
Now, simplify and solve for 'x':
1.110x = 0.809x - 40.45
0.301x = 40.45
x = 40.45 / 0.301
x ≈ 134.487 feet
Now that we have the value of 'x', we can substitute it back into equation (1) to find the height of the building:
height of the building = 134.487 * tan(39 degrees)
height of the building ≈ 134.487 * 0.809
height of the building ≈ 108.755 feet
Therefore, the height of the building is approximately 108.755 feet.