At 2 points 95 ft apart on a horizontal line perpendicular to the front of a building, the angles of elevation of the top of the building are 25 degrees and 16 degrees. How tall is the building?
Let the two points be A, and B 95 feet apart, with B closer to the building.
Let the line join the building's base at P, and the top of the building be Q.
Let the horizontal distance BP=x, and
the vertical distance
= height of building
= h
So
BPQ is a right-triangle, with QBP=25°, and ∠BPQ 90°.
So the
h = xtan(25°) .... (1)
But AP=AB+BP=95'+x
so
(95'+x)tan(16°) = h ...(2)
Solve for x and h to get:
x=151.7'
h=70.7'
Check my work.
To find the height of the building, we can use the concept of trigonometry, specifically the tangent function.
Let's label the points on the horizontal line as A and B. We can consider the top of the building as point C.
Given that the distance between points A and B is 95 ft, and the angles of elevation at points A and B are 25 degrees and 16 degrees respectively, our goal is to find the height of the building, which is the length of segment BC.
We can start by drawing a diagram of the situation. The building will be represented by a vertical line, and lines from points A and B will extend upwards to meet the top of the building at points D and E respectively.
Using trigonometry, we know that the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
In triangle ACD, let's call the height of the building (segment BC) as h, and the distance from point C to A as x. Using the tangent function for angle 25 degrees, we have:
tan(25) = h / x
Similarly, in triangle BCE, using the tangent function for angle 16 degrees, we have:
tan(16) = h / (95 - x)
We now have two equations with two unknowns (h and x).
Let's solve the system of equations to find the value of h.
First, rearrange the equations to solve for h:
h = x * tan(25) (equation 1)
h = (95 - x) * tan(16) (equation 2)
Now, we can set equation 1 equal to equation 2 and solve for x:
x * tan(25) = (95 - x) * tan(16)
Next, we can simplify the equation:
x * tan(25) = 95 * tan(16) - x * tan(16)
Add x * tan(16) to both sides:
x * tan(25) + x * tan(16) = 95 * tan(16)
Factor out x:
x * (tan(25) + tan(16)) = 95 * tan(16)
Divide both sides by (tan(25) + tan(16)):
x = (95 * tan(16)) / (tan(25) + tan(16))
Now that we have the value of x, we can substitute it back into equation 1 to find h:
h = x * tan(25)
After calculating the value of h using the formula, we will get the height of the building.