From the top of a building 21.0 m tall, the angle of elevation of the top of a taller building is 46. The angle
of depression of the base of the taller building is 51. What is the height of the taller building? (47.9 m)
if d is distance between buildings:
tan 51 = 21/d
so
d = 21 / tan 51
if h is the ADDITIONAL height of big building
tan 46 = h/d
h = d tan 46
h = (21/tan 51) tan 46
total height = 21 + h
To find the height of the taller building, we can use the trigonometric ratios of sine and tangent.
Let's begin with the angle of elevation. We have a right triangle formed by the top of the smaller building (point A), the top of the taller building (point B), and the base of the taller building (point C). The angle of elevation is the angle between the horizontal line AC and the line AB.
Using the sine ratio, we can set up the following equation:
sin(angle of elevation) = opposite/hypotenuse
sin(46°) = AB/21.0 m
To find AB, we can rearrange the equation:
AB = sin(46°) * 21.0 m
Now let's move on to the angle of depression. We have a right triangle formed by the top of the taller building (point B), the base of the taller building (point C), and the bottom of the taller building (point D). The angle of depression is the angle between the horizontal line BC and the line BD.
Using the tangent ratio, we can set up the following equation:
tan(angle of depression) = opposite/adjacent
tan(51°) = AB/BC
To find BC, we can rearrange the equation:
BC = AB/tan(51°)
Now we can substitute the value of AB from the first equation into the second equation:
BC = (sin(46°) * 21.0 m) / tan(51°)
Simplifying the equation gives us:
BC = 47.9 m
Therefore, the height of the taller building is 47.9 m.
To find the height of the taller building, we can use trigonometry. Specifically, we can use the tangent function since we have the angle of elevation and the distance between the two buildings.
Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram of the situation, labeling the given values. In this case, draw a vertical line to represent the shorter building (21.0 m) and a taller line extending from the top of the shorter building to represent the taller building (which we need to find the height of).
Step 2: Identify the relevant angles
In the diagram, we have two angles: the angle of elevation (46 degrees) and the angle of depression (51 degrees). These angles are formed by the line of sight from the person on the shorter building to the top of the taller building and from the person on the taller building to the base of the shorter building, respectively.
Step 3: Set up the trigonometric equation
Using the tangent function, we can relate the height of the taller building to the given angles and the distance between the two buildings:
tan(angle of elevation) = height of the taller building / distance between the buildings
We know the angle of elevation (46 degrees) and the distance between the buildings can be calculated by subtracting the height of the shorter building (21.0 m) from the line of sight to the top of the taller building. Let's call this distance "x".
Step 4: Solve for the distance between the buildings
To find "x," we can use the tangent function again:
tan(angle of depression) = height of the taller building / x
Substituting the given angle of depression (51 degrees) and rearranging the equation, we can solve for "x":
x = height of the taller building / tan(angle of depression)
Step 5: Substitute and solve
Now we have two equations: one relating the height of the taller building to the distance between the buildings using the angle of elevation and another equation relating the height of the taller building to the distance between the buildings using the angle of depression.
height of the taller building / x = tan(angle of elevation)
height of the taller building / x = tan(angle of depression)
Substituting the values, we have:
height of the taller building / x = tan(46 degrees)
height of the taller building / x = tan(51 degrees)
Plug in the values and solve for the height of the taller building:
height of the taller building = x * tan(46 degrees)
height of the taller building = x * tan(51 degrees)
Using a calculator, compute the values:
height of the taller building = x * 1.03553031
height of the taller building = 47.9 m
Therefore, the height of the taller building is 47.9 meters.