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A ladder against a vertical wall makes an angle of 45º with the ground. The foot of the ladder is 3m from the wall. Find the length of the ladder.

A.

2.23 m

B.

4.23 m

C.

6.23 m

D.

8.23 m

Answer with explanation

Answer: Option BExplanation

Let AB be the wall and CB, the ladder.

Then, AC = 3m and ∠ACB = 45º

∴ Length of the ladder = CB = 3 √2

= (3 × 1.41) m = 4.23 m

Workspace

From the top of a temple near a river the angles of depression of both the banks of river are 45° & 30°. If the height of the temple is 100 m then find out the width of the river.

A.

50(√3-1)m

B.

100(√3-1)m

C.

200(√3-1)m

D.

300(√3-1)m

Answer with explanation

Answer: Option BExplanation

tan 45° = AB/BD

1 = 100/BD

BD = 100

tan 30 ° = AB/BC

1/√3 = 100/BC

BC = 100 √3

Width of the river , CD = BC – BD = 100 (√3-1)

When height of tower is 1 m then width of river is √3-1Since height of tower is 100 m

Therefore ,Width of river is 100(√3-1)m

Workspace

From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. Find the height of the tower.

A.

20 m

B.

50 m

C.

130 m

D.

170 m

Answer with explanation

Answer: Option BExplanation

Let AB be the cliff and CD be the tower.

Then, AB = 25 m. From B draw BE ⊥ CD.

Let ∠EBD = ∠ACB = α.

[ ∵ BE = AC]

∴ CD = CE + DE = AB + AB = 2AB = 50m

Workspace

From the top of a building 30 m high, the top and bottom of a tower are observed to have angles of depression 30º and 45º respectively. Find the height of the tower.

A.

60(1-1/√3) m

B.

50(1-1/√3) m

C.

40(1-1/√3) m

D.

30(1-1/√3) m

Answer with explanation

Answer: Option DExplanation

Let AB be the building and CD be the tower.

Then, AB = 30 m. Let DC = x.

Draw DE ⊥ AB. Then AE = CD = x.

∴ BE = (30 – x) m.

∴ DE = AC = 30 m.

Workspace

If the angle of elevation of cloud from a point 200 m above a lake is 30º and the angle of depression of its reflection in the lake is 60º, then find the height of the cloud above the lake.

A.

400 m

B.

600 m

C.

800 m

D.

1000 m

Answer with explanation

Answer: Option AExplanation

Let C be the cloud and C’ be its reflection in the lake.

Let CS = C’S = x.

∴ CS = 400 m.

Workspace

From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is 30º. The angle of elevation of the top of a water tank (on the top of the tower) is 45º

A.

22.5, 17.4 m

B.

23.1, 16.9 m

C.

24.1, 15.7 m

D.

25.1, 14.2 m

Answer with explanation

Answer: Option BExplanation

Let BC be the tower of height h metre and CD be the water tank of height h1 metre.

Let A be a point on the ground at a distance of 40 m away from the foot B of the tower.

In ∆ABD, we have tan 45º =

In ∆ABC, we have

Substituting the value of h in (i), we get

23.1 + h1 = 40

⇒ h1 = (40 – 23.1)m = 16.9 m

Workspace

A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60º. When he moves 50 m away from the bank, he finds the angle of elevation to be 30º.

A.

8.3, 19 m

B.

45.3, 32 m

C.

41.3, 28 m

D.

43.3, 25 m

Answer with explanation

Answer: Option DExplanation

Fig.

Let AD be the tree of height h,

In ΔADC,

tan60º = h/CD

Or, √3 = h/CD

Or, CD = h/√3

In ΔADB,

tan30º = h/BD

Or, 1/√3 = h/BD

Or, BD = h√3

BD – CD = 50

h√3 – h/√3 = 50

(3h – h)/√3 = 50

2h = 50√3

Height of the tree,

h = 50√3/2

= 25√3 25×1.732

= 43.3 m.

Width of the river,

CD = h/√3

= 25√3/√3

= 25 m.

Workspace

At the foot of a mountain the elevation of its summit is 45º; after ascending 1000 m towards the mountain up a slope of 30º inclination is found to be 60º. Find the height of the mountain.

A.

1.366 km

B.

1.477 m

C.

1.788 km

D.

1.922 km

Answer with explanation

Answer: Option AExplanation

Let F be the foot and S be the summit of the mountain FOS. Then, ∠OFS = 45º and therefore ∠OSF = 45º. Consequently,

OF = OS = h km(say).

Let FP = 1000 m = 1 km be the slope so that

∠OFP = 30º. Draw PM ⊥OS and PL ⊥OF.

Join PS. It is given that ∠MPS = 60º.

In ∆FPL, We have

Now, h = OS = OF = OL + LF

In ∆PSM, we have

⇒ SM = PM. tan 60º …..(ii)

[Using equations (i) and (ii)]

⇒ h(√3 – 1) = 1

Hence, the height of the mountain is 1.366 km.

Workspace

The shadow of a vertical tower on a level ground increases by 10 m when the altitude of the sun changes from 45º to 30º. Find the height of the tower, correct to two decimal places.

A.

12.44 m

B.

13.66 m

C.

14.77 m

D.

15.22 m

Answer with explanation

Answer: Option BExplanation

Let the height of the tower AB be h and length of the shadow initially BC = y.

In Δ ABC, tan 45º = AB/BC

Or, 1 = h/y

Or, y = h —————— (i)

In Δ ABD, tan30º = AB/BD

Or, 1/√3 = h/(y + 10)

Or, y + 10 = h√3 ————— (ii)

Putting y = h in equation (ii) h + 10 = h√3

Or, h (√3 – 1) = 10

Or, h = 10/(√3 – 1) = {10 (√3 + 1)}/{(√3 – 1)(√3 + 1)}

= 10 (√3 + 1)/2

= 5(1.732 + 1)

= 5×2.732 = 13.66 m.

Workspace

An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?

A.

24 m

B.

26 m

C.

28 m

D.

30 m

Answer with explanation

Answer: Option DExplanation

Here, AB is the chimney, CD the observer and ∠ ADE the angle of elevation .

In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney.

We have AB = AE + BE = AE + 1.5

and DE = CB = 28.5 m

To determine AE, we choose a trigonometric ratio, which involves both AE and DE.

Let us choose the tangent of the angle of elevation.

Now, tan 45° = AE / DE

i.e., 1 = AE / 28.5

Therefore,

AE = 28.5

So the height of the chimney (AB) = (28.5 + 1.5) m = 30 m.

Workspace

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

A.

10√3 m

B.

20√3 m

C.

30√3 m

D.

40√3 m

Answer with explanation

Answer: Option BExplanation

AB is the tower and BC is the length of the shadow when the Sun’s altitude is 60°,

i.e., the angle of elevation of the top of the tower from the tip of the shadow is 60° and DB is the length of the shadow, when the angle of elevation is 30°.

Now, let AB be h m and BC be x m.

According to the question, DB is 40 m longer than BC.

So, DB = (40 + x) m

Now, we have two right triangles ABC and ABD.

In Δ ABC,

tan 60° = AB / BC

or, √3 = h / x ….. (1)

In Δ ABD,

tan 30° = AB / BD

i.e., 1 / √3 = h / (40 + x ) ….. (2)

From (1), we have

h = x√3

Putting this value in (2), we get

(x√3)√3 = x + 40,

i.e., 3x = x + 40

i.e., x = 20

Workspace

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100

A.

173 m

B.

200 m

C.

273 m

D.

300 m

Answer with explanation

Answer: Option CExplanation

Let AB be the lighthouse and C and D be the positions of the ships

.Then, AB = 100 m, ACB = 30° and ADB = 45°

.AB= tan 30° =1 AC = AB x 3 = 1003 m.AC3AB= tan 45° = 1 AD = AB = 100 m.AD CD = (AC + AD)= (1003 + 100) m= 100(3 + 1)= (100 x 2.73) m= 273 m.

Workspace

From a point C on a level ground, the angle of elevation of the top of a tower is 30 degree. If the tower is 100 meter high, find the distance from point C to the foot of the tower.

A.

170 meter

B.

172 meter

C.

173 meter

D.

167 meter

Answer with explanation

Answer: Option CExplanation

Let AB be the tower.

then∠ACB=30∘AB=100meterABAC=tan30∘=>100AC=13=>AC=3∗100=>AC=1.73∗100=>AC=173m

Please always remember the value of square root 3 is 1.73, and value of square root 2 is 1.41.

This will be very helpful while solving height and distance questions and saving your time.

Workspace

The angle of elevation of the sun is 60°. Find the length of the shadow of a man who is 180 cm tall

A.

127.27 cm

B.

103.92 cm

C.

311.77 cm

D.

None of these

Answer with explanation

Answer: Option BExplanation

Let AB be the man and BC be his shadow

AB = 180

tan60° = AB/BC

v3 = 180/BC

BC = 103.92 cm

Workspace

40 feet rope is cut into 2. One piece is 18 feet longer than the other. What is the length of the shorter piece?

A.

11

B.

12

C.

18

D.

22

Answer with explanation

Answer: Option AExplanation

We can also test the answer choices

Start with the middle answer choice C

C) 18

If the shorter piece is 18 feet long, then the longer piece is 36 feet long

Sum of lengths = 18 + 36 = 54

NO GOOD. We want a sum of 40 feet (as per the question)

So, we can ELIMINATE C.

Also, since we need the two pieces to be SHORTER, we can also ELIMINATE D and E

Now try answer choice B

B) 11

If the shorter piece is 11 feet long, then the longer piece is 29 feet long

Sum of lengths = 11 + 29 = 40

PERFECT!!!

Answer: B

Workspace

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