Height and Distance

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This topic has many practical application in day to day life. In engineering stage it is used in surveying. The basic purpose is to find the unknown variables by observing the angle of the line of sight. This is done by using some the fact that in a right angled triangle the ratio of any two sides is a function of the angle between them. From exam point of view this is one of the more tough sections and tedious to some extent. So an aspirant must thoroughly solve all the questions given here.

Important Formulas

1. Trigonometric Basics

sinθ=oppositeside/hypotenuse=y/r

cosθ=adjacentside/hypotenuse=x/r

tanθ=oppositeside/adjacentside=y/x

cosecθ=hypotenuse/oppositeside=r/y

secθ=hypotenuse/adjacentside=r/x

cotθ=adjacentside/oppositeside=x/y

From Pythagorean theorem, x2+y2=r2 for the right angled triangle mentioned above

2. Basic Trigonometric Values

3. Trigonometric Formulas

Degrees to Radians and vice versa

360°=2π radian

Trigonometry – Quotient Formulas

tanθ=sinθ/cosθ

cotθ=cosθ/sinθ

Trigonometry – Reciprocal Formulas

cosecθ=1/sinθ

secθ=1/cosθ

cotθ=1/tanθ

Trigonometry – Pythagorean Formulas

sin2θ+cos2θ=1

sec2θ−tan2θ=1

cosec2θ−cot2θ=1

4. Angle of Elevation

Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, angle of elevation is the angle between the horizontal and the line from the object to the observer’s eye (the line of sight).

i.e., angle of elevation =  AOP

5. Angle of Depression

Suppose a man from a point O looks down at an object P, placed below the level of his eye. Then, angle of depression is the angle between the horizontal and the observer’s line of sight

i.e., angle of depression =  AOP

6. Angle Bisector Theorem

Consider a triangle ABC as shown above. Let the angle bisector of angle A intersect side BC at a point D. Then BD/DC=AB/AC

(Note that an angle bisector divides the angle into two angles with equal measures.
i.e., BAD = CAD in the above diagram)

7. Few Important Values to memorize

√2=1.414, √3=1.732, √5=2.236

Solved Examples

Level 1

Answer : Option D

Explanation :

Consider the diagram shown above where PR represents the ladder and RQ represents the wall.

cos 60° = PQ/PR

1/2=12.4/PR

PR=2×12.4=24.8 m

Answer : Option A

Explanation :

tan 30°=RQ/PQ

1/√3=200/PQ

PQ=200√3=200×1.73=346 m

Answer : Option C

Explanation :

Consider the diagram shown above where QR represents the tree and PQ represents its shadow

We have, QR = PQ
Let QPR = θ

tan θ = QR/PQ=1 (since QR = PQ)

=> θ = 45°

i.e., required angle of elevation = 45°

Answer : Option B

Explanation :

SR = PQ = 2 m

PS = QR = 10√3m

tan 30°=TS/PS

1/3=TS/10√3

TS=10√3/√3=10 m

TR = TS + SR = 10 + 2 = 12 m

Answer : Option B

Explanation :

Let AC be the tower and B be the position of the bus.

Then BC = the distance of the bus from the foot of the tower.

Given that height of the tower, AC = 80 m and the angle of depression, DAB = 30°

ABC = DAB = 30° (Because DA || BC)

tan 30°=AC/BC=>tan 30°=80/BC=>BC = 80/tan 30°=80/(1/√3)=80×1.73=138.4 m

i.e., Distance of the bus from the foot of the tower = 138.4 m

Answer : Option B

Explanation :

Let RQ be the pole and PQ be the shadow

Given that RQ = 18 m and PQ = 6√3 m

Let the angle of elevation, RPQ = θ

From the right  PQR,

tanθ=RQ/PQ=18/6√3=3/√3=(3×√3)/( √3×√3)=3√3/3=√3

θ=tan−1(3√)=60°

Level 2

Answer : Option B

Explanation :

Consider the diagram shown above. Let AB be the tower. Let D and C be the positions of the car

Then, ADC = 30° , ACB = 45°

Let AB = h, BC = x, CD = y

tan 45°=AB/BC=h/x

=>1=h/x=>h=x——(1)

tan 30°=AB/BD=AB/(BC + CD)=h/(x+y)

=>1/√3=h/(x+y)

=>x + y = √3h

=>y = √3h – x

=>y = √3h−h(∵ Substituted the value of x from equation 1 )

=>y = h(√3−1)

Given that distance y is covered in 8 minutes
i.e, distance h(√3−1) is covered in 8 minutes

Time to travel distance x
= Time to travel distance h (∵ Since x = h as per equation 1).

Let distance h is covered in t minutes

since distance is proportional to the time when the speed is constant, we have

h(√3−1)∝8—(A)

h∝t—(B)

(A)/(B)=>h(√3−1)/h=8/t

=>(√3−1)=8/t

=>t=8/(√3−1)=8/(1.73−1)=8/.73=800/73minutes ≈10 minutes 57 seconds

Answer : Option C

Explanation :

Consider the diagram shown above. AC represents the tower and DE represents the pole

Given that AC = 15 m , ADB = 30°, AEC = 60°

Let DE = h

Then, BC = DE = h, AB = (15-h) (∵ AC=15 and BC = h), BD = CE

tan 60°=AC/CE=>√3=15/CE=>CE = 15√3— (1)

tan 30°=AB/BD=>1/√3=(15−h)/BD

=>1/√3=(15−h)/(15/√3)(∵ BD = CE and Substituted the value of CE from equation 1)

=>(15−h)=(1/√3)×(15/√3)=15/3=5

=>h=15−5=10 m

i.e., height of the electric pole = 10 m

Answer : Option C

Explanation :

Let BD be the lighthouse and A and C be the positions of the ships.
Then, BD = 100 m,  BAD = 30° ,  BCD = 45°

tan 30° = BD/BA⇒1/√3=100/BA

⇒BA=100√3

tan 45° = BD/BC

⇒1=100/BC

⇒BC=100

Distance between the two ships = AC = BA + BC
=100√3+100=100(√3+1)=100(1.73+1)=100×2.73=273 m

Answer : Option C

Explanation :

Consider the diagram shown above. AC represents the hill and DE represents the pole

Given that AC = 100 m

XAD = ADB = 30° (∵ AX || BD )
XAE = AEC = 60° (∵ AX || CE)

Let DE = h

Then, BC = DE = h, AB = (100-h) (∵ AC=100 and BC = h), BD = CE

tan 60°=AC/CE

=>√3=100/CE=>CE = 100/√3— (1)

tan 30°=AB/BD=>1/√3=(100−h)/BD

=>1/√3=(100−h)/(100/√3)(∵ BD = CE and Substituted the value of CE from equation 1 )

=>(100−h)=1/√3×100/√3=100/3=33.33=>h=100−33.33=66.67 m

i.e., the height of the pole = 66.67 m

Answer : Option A

Explanation :

Let DC be the vertical tower and AD be the vertical flagpole. Let B be the point of observation.

Given that AD = 18 m, ABC = 60°, DBC = 30°

Let DC be h.

tan 30°=DC/BC

1/√3=h/BC

h=BC√3—— (1)

tan 60°=AC/BC

√3=(18+h)/BC

18+h=BC×√3—— (2)

(1)/(2)=>h/(18+h)=(BC/√3)/(BC×√3)=1/3

=>3h=18+h=>2h=18=>h=9 m

i.e., the height of the tower = 9 m

Answer : Option B

Explanation :

Let C be the position of John. Let A be the position at which balloon leaves the earth and B be the position of the balloon after 2 minutes.

Given that CA = 150 m, BCA = 60°

tan 60°=BA/CA

√3=BA/150

BA=150√3

i.e, the distance travelled by the balloon = 150√3meters

time taken = 2 min = 2 × 60 = 120 seconds

Speed = Distance/Time=150√3/120=1.25√3=1.25×1.73=2.16 meter/second

Answer : Option B

Explanation :

Let DC be the wall, AB be the tree.

Given that DBC = 30°, DAE = 60°, DC = 11 m

tan 30°=DC/BC

1/√3=11/BC

BC = 11√3 m

AE = BC =11√3 m—— (1)

tan 60°=ED/AE

√3=ED/11√3[∵ Substituted the value of AE from (1)]

ED =11√3×√3=11×3=33

Height of the tree = AB = EC = (ED + DC) = (33 + 11) = 44 m

Answer : Option B

Explanation :

Let AB and CD be the poles with heights h and 2h respectively

Given that distance between the poles, BD = 200 m

Let E be the middle point of BD.

Let AEB = θ and CED = (90-θ) (∵ given that angular elevations are complementary)

Since E is the middle point of BD, we have BE = ED = 100 m

From the right  ABE,
tanθ=AB/BE and tanθ=h/100

h = 100tanθ—— (1)

From the right  EDC,

tan(90−θ)=CD/ED

cotθ=2h/100[∵tan(90−θ)=cotθ]

2h =100cotθ—— (2)

(1) × (2) => 2h2=1002[∵tanθ×cotθ=tanθ×1/tanθ=1]

=>√2h=100

=>h=100/√2=(100×√2)/( √2×√2)=50√2=50×1.41=70.5

2h=2×70.5=141

i.e., the height of the poles are 70.5 m and 141 m.

Answer : Option D

Explanation :

Let AB be the man and CD be the window

Given that the height of the man, AB = 180 cm, the distance between the man and the wall, BE = 5 m,
DAF = 45° , CAF = 60°

From the diagram, AF = BE = 5 m

From the right  AFD, tan45°=DF/AF

1=DF/5

DF = 5—— (1)From the right  AFC, tan60°=CF/AF

√3=CF/5

CF=5√3—— (2)

Length of the window = CD = (CF – DF)

=5√3−5[∵ Substitued the value of CF and DF from (1) and (2)]=5(√3−1)=5(1.73−1)=5×0.73=3.65 m

Answer : Option D

Explanation :

Let A be the foot and C be the summit of a mountain.

Given that CAB = 45°

From the diagram, CB is the height of the mountain. Let CB = x

Let D be the point after ascending 2 km towards the mountain such that
AD = 2 km and given that DAY = 30°

It is also given that from the point D, the elevation is 60°

i.e., CDE = 60°

From the right  ABC,

tan45°=CB/AB

=>1=x/AB[∵ CB = x (the height of the mountain)]

=>AB = x—— (eq:1)

From the right  AYD,

sin30°=DY/AD

=>1/2=DY/2(∵ Given that AD = 2)

=> DY=1—— (eq:2)

cos30°=AY/AD=>√3/2=AY/2(∵ Given that AD = 2)=> AY=√3—— (eq:3)

From the right  CED, tan60°=CE/DE=>tan60°=(CB – EB)/YB∵ [CE = (CB – EB) and DE = YB)]

=>tan60°=(CB – DY)/(AB – AY)[ ∵ EB = DY and YB = (AB – AY)]

=>tan60°=(x – 1)/(x -√3)∵ [CB = x, DY = 1(eq:2), AB=x (eq:1) and AY = 3√(eq:3)]

=>√3=(x – 1)/(x -√3)=>x√3−3=x−1=>x(√3−1)=2=>0.73x=2=>x=2/0.73=2.7

i.e., the height of the mountain = 2.7 km,

Height and Distance are two important concepts in geometry and physics. Height is the vertical distance from a point to a reference point, while distance is the length of the line segment between two points.

Absolute height is the height of a point above sea level. It is measured in meters or feet. Barometric height is the height of a point above the mean sea level, as determined by the barometric pressure. Celestial height is the angle between the horizon and a celestial object, such as a star or planet. Cumulative distribution function is a function that gives the Probability that a random variable will be less than or equal to a certain value. Depth is the vertical distance from a point to the surface of a body of water. Distance is the length of the line segment between two points. Earth’s radius is the distance from the center of the Earth to its surface. Foot is a unit of length equal to 0.3048 meters. Geodetic height is the height of a point above the geoid, which is an idealized, smooth surface that approximates the Earth’s surface. Geoid is the shape of the Earth’s surface if it were not affected by Mountains, valleys, or other topographic features. Height above sea level is the vertical distance from a point to the mean sea level. Height of a mountain is the vertical distance from the base of the mountain to its summit. Hour angle is the angle between the hour circle of a celestial object and the observer’s meridian. Light year is the distance that light travels in one year, which is approximately 9.46 trillion kilometers. Lunar distance is the distance between the Earth and the Moon. Nautical mile is a unit of length equal to 1.852 kilometers. Orthometric height is the height of a point above the ellipsoid, which is a mathematical model of the Earth’s surface. Parallax is the apparent displacement of an object due to a change in the observer’s position. Plane table surveying is a method of surveying that uses a plane table and a alidade to measure angles and distances. Radar altimeter is an instrument that uses radar to measure the height of an aircraft above the ground. Ranging is the process of determining the Distance Between Two Points. Sextant is an instrument used to measure angles between celestial objects. Stadia surveying is a method of surveying that uses stadia rods to measure distances. Surveying is the science of determining the position of points on the Earth’s surface. Topography is the study of the Earth’s surface features. Vertical datum is a reference surface used to measure heights. Vertical exaggeration is a technique used to make topographic maps more visually appealing. Zenith distance is the angle between the zenith and a celestial object.

These are just a few of the many concepts related to height and distance. Height and distance are important concepts in many fields, including geometry, physics, surveying, and navigation.

Sure, here are some frequently asked questions and short answers about other topics:

• What is the difference between mass and weight?

Mass is a measure of how much matter is in an object, while weight is a measure of the force of gravity on that object. Mass is always the same, no matter where you are, but weight can change depending on the strength of the gravitational field.

• What is the speed of light?

The speed of light is the speed at which light travels in a vacuum. It is a fundamental constant of nature and is the fastest speed at which information can travel. The speed of light is 299,792,458 meters per second.

• What is the definition of time?

Time is a fundamental part of the universe and is used to measure the interval between events. Time is often thought of as a linear progression of events, but it can also be thought of as a cyclical or even chaotic process.

• What is the meaning of life?

The meaning of life is a question that has been pondered by philosophers and theologians for centuries. There is no one answer that will satisfy everyone, but some possible answers include finding happiness, making a difference in the world, or simply living each day to the fullest.

• What is the universe?

The universe is everything that exists, including all matter and energy, as well as space and time. The universe is thought to have begun with the Big Bang about 13.8 billion years ago.

• What is consciousness?

Consciousness is the state of being aware of and responsive to one’s surroundings. It is a complex phenomenon that is still not fully understood by scientists. Some possible explanations for consciousness include the mind-body problem, the hard problem of consciousness, and the problem of qualia.

• What is love?

Love is a complex emotion that is often described as a mixture of many other emotions, such as happiness, sadness, anger, and fear. Love can be felt for a person, place, thing, or idea. It is a powerful emotion that can have a profound impact on our lives.

• What is death?

Death is the end of life. It is a natural process that will happen to everyone eventually. Death can be caused by a variety of factors, such as old age, disease, or injury. It is a difficult and often painful experience, but it is also a natural part of life.

• What is God?

God is a term used to describe a variety of different concepts, including a supreme being, a universal spirit, or a higher power. There is no one definition of God that is universally accepted. Some people believe in God, while others do not.

• What is the purpose of life?

The purpose of life is a question that has been pondered by philosophers and theologians for centuries. There is no one answer that will satisfy everyone, but some possible answers include finding happiness, making a difference in the world, or simply living each day to the fullest.

• What happens after death?

What happens after death is a question that has been pondered by people for centuries. There is no one answer that is universally accepted. Some people believe that there is an afterlife, while others believe that death is the end of existence.

• What is the meaning of the universe?

The meaning of the universe is a question that has been pondered by philosophers and theologians for centuries. There is no one answer that will satisfy everyone, but some possible answers include the search for truth, the pursuit of knowledge, or the experience of beauty.

Sure, here are some MCQs without mentioning the topic Height and Distance:

1. A car travels 100 miles in 2 hours. What is the Average speed of the car?
2. A train travels 60 miles in 1 hour. What is the average speed of the train?
3. A plane travels 500 miles in 1 hour. What is the average speed of the plane?
4. A boat travels 20 miles in 1 hour. What is the average speed of the boat?
5. A bicycle travels 10 miles in 1 hour. What is the average speed of the bicycle?
6. A person walks 2 miles in 1 hour. What is the average speed of the person?
7. A snail crawls 100 feet in 1 hour. What is the average speed of the snail?
8. A cheetah runs 70 miles per hour. What is the average speed of the cheetah?
9. A turtle crawls 0.1 miles per hour. What is the average speed of the turtle?
10. A fly buzzes at 10 miles per hour. What is the average speed of the fly?

I hope these MCQs are helpful!