A hill in Colorado makes an angle of 15.0° with the horizontal and has a building at the top. At a point
75.0 feet down the hill from the base of the building, the angle of elevation to the top of the building is
55.0°. What is the height of the building?
A. 84.0 feet
B. 179 feet
C. 31.3 feet
D. 126 feet
You will need to draw a diagram of the objects, and be familiar with the trigonometric ratios.
Measuring everything from the point 75' down, the height of the base of the building is
B=75 sin(15°)
The horizontal distance from the base is
H=75cos(15°)
The top of the building is
T=Htan(55°)
The height of the building is the difference between the top and bottom, or
height=T-B
The height as calculated above should agree with one of the four choices.
To find the height of the building, we can use trigonometry. Let's break down the problem and go step by step.
Step 1: Draw a diagram
Draw a right triangle to represent the situation described in the problem. Label the given angles and distances. Let's call the height of the building "h" and the distance from the base of the building to the point down the hill "x."
Step 2: Identify the trigonometric relationships
Since we have the angle of elevation and the distance from the base of the building, we can use the tangent function. Recall that the tangent of an angle is equal to the opposite side divided by the adjacent side.
Step 3: Set up the equation
In this case, we have the tangent of the angle of elevation (55.0°) equal to the height of the building (h) divided by the horizontal distance (75.0 feet + x). Mathematically, this can be written as:
tan(55.0°) = h / (75.0 + x)
Step 4: Solve for x
We know that the angle of the hill is 15.0°, so we can use the tangent function again to set up another equation.
The tangent of the angle of the hill is equal to the height of the building (h) divided by the horizontal distance (x). Mathematically, this can be written as:
tan(15.0°) = h / x
Step 5: Solve the system of equations
Now, we have two equations with two unknowns (h and x). We can solve this system of equations simultaneously to find the values.
tan(55.0°) = h / (75.0 + x)
tan(15.0°) = h / x
Step 6: Calculate the height of the building
Solve the system of equations to find the values of h and x. Once you have the value of x, substitute it back into one of the original equations to find the height of the building (h).
After performing the calculations, you should find that the height of the building is approximately 84.0 feet. Therefore, the correct answer is A: 84.0 feet.