From a certain point, the angle of elevation of the top of the building is 38. From a point 75 feet nearer the building, the angle of elevation is 65. Find the height of the building.
note that
h cot38° - h cot75° = 65
92.2 feet
To find the height of the building, we will use the tangent function.
Let's label the height of the building as h.
From the first point, the angle of elevation is 38 degrees. This means that the tangent of the angle is equal to the height of the building divided by the distance from the building to the point. Let's call this distance x.
So we have:
tan(38) = h / x
Next, from the second point, which is 75 feet closer to the building, we have an angle of elevation of 65 degrees. This time, the distance from the building to the point is x - 75.
So we have:
tan(65) = h / (x - 75)
Now we have two equations with two unknowns. Let's solve them simultaneously.
Rearrange the first equation:
h = x * tan(38)
Substitute this into the second equation:
tan(65) = (x * tan(38)) / (x - 75)
Now we can simplify and solve for x.
Multiply both sides of the equation by (x - 75):
(x - 75) * tan(65) = x * tan(38)
Expand and rearrange the equation:
xtan(65) - 75tan(65) = xtan(38)
xtan(65) - xtan(38) = 75tan(65)
Factor out x:
x(tan(65) - tan(38)) = 75tan(65)
Divide both sides by (tan(65) - tan(38)):
x = (75tan(65)) / (tan(65) - tan(38))
Now we can substitute this value of x back into the first equation to find the height of the building:
h = x * tan(38)
h = ((75tan(65)) / (tan(65) - tan(38))) * tan(38)
Use a calculator to calculate these values:
x ≈ 134.32 feet
h ≈ 88.88 feet
Therefore, the height of the building is approximately 88.88 feet.
To find the height of the building, we can use the trigonometric concept of tangent.
Let's denote the height of the building as 'h' feet.
We are given two angles and the distance between the two points. We can set up two right triangles to represent these scenarios.
Right triangle 1:
From the initial point, the angle of elevation to the top of the building is 38 degrees. Let's denote the distance from this point to the building as 'x' feet.
In this triangle, the opposite side is 'h' feet (height of the building) and the adjacent side is 'x' feet (distance from the initial point).
Using the tangent function, we have:
tan(38) = h / x
Right triangle 2:
From the point 75 feet nearer to the building, the angle of elevation to the top of the building is 65 degrees. The distance from this point to the building is 'x + 75' feet.
In this triangle, the opposite side is 'h' feet (height of the building) and the adjacent side is 'x + 75' feet (distance from the second point).
Using the tangent function again, we have:
tan(65) = h / (x + 75)
Now, we have two equations involving 'h', 'x', and 'x + 75'.
By solving the system of equations, we can find the values of 'h' and 'x'.
Once we have the value of 'h', we can determine the height of the building.