Download 3_Object Heights Power Point - Maine-Math-in-CTE

How High Is That Building?
Can anyone think of an object on the school campus that
we could not use a tape measure to directly measure?
Does anyone have any idea how we can find the height of an
object we can’t directly measure?
?
If we use a right triangle to find the height of the object, which side
of the triangle can we directly measure?
?
In our real life example, what distance is represented by the base of the
triangle?
In our real life example, what distance is represented by the base of the
triangle?
d
The distance from where we are standing
to the building
What kind of instrument can we use to measure the acute angle at the
base of the triangle?
What kind of instrument can we use to measure the acute angle at the
base of the triangle?
Which trigonometric function can we use to find the length of the
opposite side if we know the length of the side adjacent to the
acute base angle and the measure of the angle?
opposite
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
sin  
Opposite Side
Known
angle
Adjacent Side
Which trigonometric function can we use if we know the length of
the side adjacent to the acute base angle and the measure of the
angle?
opposite
tan  
adjacent
Opposite
Side
Known
angle
Adjacent Side
Find the height of the building if the distance from the building
is 45 feet and angle of elevation is 27o .
Find the height of the building if the distance from the building
is 45 feet and angle of elevation is 27o .
opposite
tan  
adjacent
h
i
tan =
d
Find the height of the building if the distance from the building
is 45 feet and angle of elevation is 27o .
h
i
tan =
d
o
tan 27 =
h
45'
Find the height of the building if the distance from the building
is 45 feet and angle of elevation is 27o .
h
i
tan =
d
o
tan 27 =
h
45'
o
45' * tan27 = h
Find the height of the building if the distance from the building
is 45 feet and angle of elevation is 27o .
h
i
tan =
d
o
tan 27 =
h
45'
o
45' * tan27 = h
22.9
feet = h
So the height of the building is 22.9 feet.
You want to drive your truck hauling an excavator under an overpass that has a
clearance of 15.5 ft? You set up your surveying equipment and measure the distance
to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the
overpass?
You want to drive your truck hauling an excavator under an overpass that has a
clearance of 15.5 ft? You set up your surveying equipment and measure the distance
to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the
overpass?
tan i =
height of
load

Distance to truck
height of load
distance to truck
You want to drive your truck hauling an excavator under an overpass that has a
clearance of 15.5 ft? You set up your surveying equipment and measure the distance
to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the
overpass?
tan i =
height of
load
o

Distance to truck
height of load
distance to truck
tan19 =
height of load
48'
You want to drive your truck hauling an excavator under an overpass that has a
clearance of 15.5 ft?. You set up your surveying equipment and measure the distance
to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the
overpass?
tan i =
height of
load
o

Distance to truck
height of load
distance to truck
tan19 =
height of load
48'
o
48' * tan19 = height of load
16.5' = height of load
The excavator will not fit under
the overpass.
You can walk across the Sydney Harbor Bridge and take a photo of the
Opera House from about the same height as top of the highest sail.
This photo was taken from a point about 500 m horizontally from the
Opera House and we observe the waterline below the highest sail as
having an angle of depression of 8°. How high above sea level is the
highest sail of the Opera House?
tan 8° = h/500
h = 500 tan 8° = 70.27 m.
So the height of the tallest point is around 70 m.
A tree casts a shadow 70 feet long at an angle of elevation of 30º.
How tall is the tree?
A tree casts a shadow 70 feet long at an angle of elevation of 30º.
How tall is the tree?
x
tan 30º 
70
70  tan 30º  x
x  40.4 ft
When building a cabin with a 4/12 pitch roof, at what angle
should the plumb cut of the rafter be cut?
When building a cabin with a 4/12 pitch roof, at what angle
should the plumb cut of the rafter be cut?
opposite
tan 
adjacent
12
tan 
4
 12 
tan    
4
  71.6
1
o
So the plumb cut will be 71.6
o
A flagpole stands in the middle of a flat, level field. Fifty feet
away from its base a surveyor measures the angle to the
top of the flagpole as 48°. How tall is the flagpole?
A flagpole stands in the middle of a flat, level field. Fifty feet
away from its base a surveyor measures the angle to the
top of the flagpole as 48°. How tall is the flagpole?
Let a denote the height of
the flagpole.
a
 tan 48
50
o
a = 50 tan 48°  55.5 ft.
Two trees stand opposite one another, at points A and B, on
opposite banks of a river.
Distance AC along one bank is perpendicular to BA, and is
measured to be 100 feet. Angle ACB is measured to be
79°. How far apart are the trees; that is, what is the width w
of the river?
Two trees stand opposite one another, at points A and B, on
opposite banks of a river.
Distance AC along one bank is perpendicular to BA, and is
measured to be 100 feet. Angle ACB is measured to be
79°. How far apart are the trees; that is, what is the width w
of the river?
w
= tan 79° ,
100
w = 100 × tan 79° = 514.5 ft
Standing across the street 50 feet from a building, the angle to
the top of the building is 40°. An antenna sits on the front edge
of the roof of the building. The angle to the top of the antenna
is 52°. How tall is the building. How tall is the antenna itself, not
including the height of the building?
Standing across the street 50 feet from a building, the angle to
the top of the building is 40°. An antenna sits on the front edge
of the roof of the building. The angle to the top of the antenna
is 52°. How tall is the building. How tall is the antenna itself, not
including the height of the building?
Let a represent the height of the building and
h the height of the antenna.
Then the following relationships hold:
a = 50 tan 40°, a + h = 50 tan 52°
50 tan 40° + h = 50 tan 52°
h = 50 ( tan 52° - tan 40° )  22 ft.
You are looking up at a fourth story window, 40 feet up in a
building. You are 100 feet away from the building, across the
street. What is the angle of elevation?
You are looking up at a fourth story window, 40 feet up in a
building. You are 100 feet away from the building, across the
street. What is the angle of elevation?
The building is perpendicular to the ground. Therefore the 40 feet
opposite the angle of elevation A and the 100 feet you are away
from the building gives us
tan i = h
d
tan i = 40
100
tan i = .4
tan- 1 (.4) = i
i = 21.8o
0.4 gives us angle A is approximately 21.8º
Now, let’s get outside and try it !