Unit 5: Height, Distance, and Analytical Reasoning - Subjective Questions

Angle of elevation: It is the angle formed between the horizontal line through the observer's eye and the line of sight when the object being observed is above the horizontal level.

Angle of depression: It is the angle formed between the horizontal line through the observer's eye and the line of sight when the object is below the horizontal level.

• Both angles are measured from a horizontal line.
• Since horizontal lines are parallel, the angle of depression from the top of an object equals the corresponding angle of elevation from the lower point.
• Height-and-distance problems are converted into right-angled triangles.
• The appropriate trigonometric ratio is then applied:

Thus, if any two of the angle, height, and horizontal distance are known, the third quantity can usually be determined.

Let the height of the tower be m and the horizontal distance from the observation point to its foot be m.

Using the tangent ratio:

Since

we get

Therefore,

Hence, the height of the tower is m.

Let the horizontal distance of the car from the building be m.

The angle of depression from the roof equals the angle of elevation from the car. Therefore, the angle of elevation is .

In the resulting right-angled triangle:

Since ,

Thus,

Therefore, the car is m from the building.

Let the height of the pole be m. The pole, its shadow, and the Sun's ray form a right-angled triangle.

Here:

• The height of the pole is the side opposite to .
• The shadow is the adjacent side.

Therefore, the tangent ratio is appropriate:

Using :

and hence

The height of the pole is m, approximately m.

Let the nearer observation point be m from the tower. The farther point is then m from the tower. Let the tower's height be m.

At the nearer point:

Therefore,

At the farther point:

Therefore,

Equating the expressions for :

Hence,

The height of the tower is m.

Let the man's distance from the tower after walking forward be m. His original distance is therefore m. Let the tower's height be m.

From the original position:

Since :

From the nearer position:

Thus,

Equating the two expressions:

The height is

The original distance is also equal to because the original angle is .

Therefore:

• Height of the tower: m
• Original distance from the tower: m

Let the horizontal distances of the boats from the foot of the lighthouse be m and m. Since the boats are on opposite sides, their total distance apart is .

For the first boat:

For the second boat:

Therefore, the distance between the boats is

Hence, the boats are m apart.

In a linear seating arrangement, people or objects are placed in a straight row. The main principles are:

• Fix the direction: If everyone faces north, a person's left and right match the observer's left and right. If everyone faces south, the person's left and right are reversed from the observer's perspective.
• Use definite positions first: Statements such as "A sits at the extreme left" or "B is third from the right" should be placed before relative clues.
• Interpret immediate neighbors carefully: If A sits immediately to the left of B, there is no person between A and B.
• Interpret relative positions: If A sits second to the left of B, exactly one seat lies between them.
• Combine linked clues: Clues involving common persons should be grouped to form blocks.
• Check all conditions: The completed arrangement must satisfy every clue, including negative conditions such as "C is not adjacent to D."

A table of numbered positions is often useful for avoiding ambiguity.

Number the positions from left to right as to .

• C is at the extreme left, so C occupies position .
• A is second to the left of D, and B is immediately to the right of A.
• Therefore, the linked block must be A-B-D because if A is at position , B is at and D is at .
• The block cannot begin at position because C is already there.
• If it begins at position , A, B, and D occupy positions , , and .
• The remaining consecutive positions and accommodate E-F.

Thus, the arrangement from left to right is:

Verification:

• A is second to the left of D.
• B is immediately right of A.
• C is at the extreme left.
• E is immediately left of F.
• D is not at an end.

There are seven positions, numbered from left to right. The middle position is position .

• S occupies position .
• P sits third to the left of S, so P occupies position .
• Q sits immediately to the right of P, so Q occupies position .
• V occupies the extreme right end, position .
• The vacant positions are , , and for R, T, and U.
• Since T sits immediately to the left of U, T and U must occupy positions and .
• R therefore occupies position .

The final arrangement is:

Position is second from the right. Therefore, U sits second from the right.

When people face north:

• Their left corresponds to the observer's left.
• Their right corresponds to the observer's right.
• If A is immediately to the left of B, the visible order is A-B.

When people face south:

• Their left corresponds to the observer's right.
• Their right corresponds to the observer's left.
• If A is immediately to the left of B, the visible order from the observer's viewpoint is B-A.

Example: Suppose two seats are shown from left to right.

• Facing north, "P is to the left of Q" gives P-Q.
• Facing south, "P is to the left of Q" gives Q-P from the observer's perspective.

Thus, the facing direction must be established before interpreting left-right clues.

Number the positions from left to right as to .

Since A and H occupy the two ends, test the placement A at position and H at position .

• B is third to the right of A, so B is at position .
• C is immediately to the left of B, so C is at position .
• D is second to the right of B, so D is at position .
• The vacant positions are , , and for E, F, and G.
• E must sit immediately to the left of F. The only available consecutive pair is positions -, -, or -, but positions , , and are occupied. Therefore this assumed end placement does not work.

Reverse the end occupants: H at position and A at position . Since B must be third to the right of A, this is also impossible when all face north.

Therefore, the given clues are inconsistent and no valid arrangement exists. The contradiction follows from combining the end condition with the fixed relative positions and the E-F adjacency requirement. In analytical reasoning, identifying such an inconsistency is a valid conclusion.

Translate each statement into an order relation:

• K is ahead of L: .
• K is behind M: .
• N is behind L: .
• N is ahead of O: .

Combining these relations gives:

Therefore, the order from front to back is:

Transitive reasoning means that if one person is ahead of a second and the second is ahead of a third, then the first is also ahead of the third. For example, since and , it follows that .

A circular arrangement differs from a linear arrangement in the following ways:

• A circle has no fixed beginning or end, whereas a row has extreme left and extreme right positions.
• Inward-facing people have their left in the clockwise direction and their right in the anticlockwise direction.
• Outward-facing people have their left in the anticlockwise direction and their right in the clockwise direction.
• Every person generally has two immediate neighbors.
• Circular arrangements that differ only by rotation are considered identical. For example, shifting every person one place clockwise does not produce a new relative arrangement.
• To solve a problem, one person may be fixed at an arbitrary reference position. This removes rotational duplication.

If reflections are treated as distinct, clockwise and anticlockwise arrangements must not be interchanged.

Fix A at a reference position. Since everyone faces the center, a person's left is clockwise and right is anticlockwise.

• B sits immediately to the left of A, so place B clockwise from A.
• C sits immediately to the left of B, so place C clockwise from B.
• D sits opposite A.
• E sits immediately to the right of A, so place E anticlockwise from A.
• The remaining seat is occupied by F.

Starting from A and moving clockwise, the arrangement is:

Verification:

• B is immediately left of A.
• C is immediately left of B.
• D is opposite A.
• E is immediately right of A.

Fix P at a reference position. For inward-facing people, left is clockwise and right is anticlockwise.

• Q is second to the left of P, so Q is placed two seats clockwise from P.
• R sits opposite P, so R is four seats from P.
• S sits immediately to the right of R, so S is one seat anticlockwise from R.
• T sits opposite Q.
• U sits immediately to the left of P, so U is one seat clockwise from P.
• V sits immediately to the right of T, so V is one seat anticlockwise from T.
• W occupies the only remaining seat.

Starting from P and moving clockwise, the arrangement is:

W is seated between T and P. Therefore, the immediate neighbors of W are T and P.

Facing the center:

• Left corresponds to the clockwise direction.
• Right corresponds to the anticlockwise direction.
• If B is immediately left of A, B is placed one seat clockwise from A.

Facing outward:

• Left corresponds to the anticlockwise direction.
• Right corresponds to the clockwise direction.
• If B is immediately left of A, B is placed one seat anticlockwise from A.

The reversal occurs because an outward-facing person's orientation is opposite to that of an inward-facing person. A reliable method is to imagine oneself sitting in the person's position and facing in the stated direction before deciding left or right.

A systematic approach is as follows:

• Fix A at any reference position because rotations are equivalent.
• Since A sits between B and C, place B and C in the two seats adjacent to A. Their exact order is not specified, so two initial cases are possible.
• D sits second to the left of A. As everyone faces the center, count two seats clockwise from A and place D there.
• E sits immediately to the right of D, so place E one seat anticlockwise from D.
• Place F only in a remaining seat that is not adjacent to A.
• Place G in the final vacant seat.
• Check every clue in each case.

More than one arrangement may remain because:

• The clue "A sits between B and C" does not specify whether B is on A's left or right.
• F is restricted only by a negative condition.
• No additional clue fixes the relative positions of F and G.

Therefore, the information may define a set of valid arrangements rather than one unique arrangement.

In a linear arrangement, different people can be arranged in

ways. In a circle, however, rotating an arrangement does not change the relative positions. Each circular arrangement is counted times in , once for each choice of the person appearing at a fixed reference position.

Therefore, the number of distinct circular arrangements is

Another way to obtain the result is to fix one person in a reference seat and arrange the remaining people around that person:

For six people:

Hence, six different people can be seated around a circular table in distinct ways when rotations are identical.

For people facing the center, the left direction is clockwise.

• B is second to the left of A: Starting from A, move two seats clockwise. The first seat is immediately left of A, and the second seat is occupied by B. Exactly one person sits between A and B.
• C is opposite A: In a circle of eight seats, the opposite seat is four positions away in either direction. Therefore, three people sit between A and C along either side of the circle.
• D is an immediate neighbor of A: D occupies either of the two seats directly adjacent to A. No person sits between A and D.

Thus, the relevant counts are:

• Second to the left: move seats.
• Opposite in an eight-person circle: move seats.
• Immediate neighbor: move seat in either direction.