Question

Write a relation between \(d\) (the height of window) and \(y\).

Answer 1

Step 1: Understanding the Geometry:
In right triangle AWX:
• AX = d (height of window from ground)
• AW = y (line-of-sight distance)
• Angle of depression = 30°

Since angle of depression equals angle of elevation,
∠AWX = 30°.

Step 2: Using Cosine Ratio (Alternative Method):
Instead of sine, we use cosine first.

cos 30° = Adjacent / Hypotenuse
= XW / y

cos 30° = √3/2

So,
√3/2 = 54 / y

y = (54 × 2) / √3
y = 108 / √3

Step 3: Now Find d Using Tangent:
tan 30° = Opposite / Adjacent
= d / 54

1/√3 = d / 54

d = 54 / √3

Step 4: Establishing the Relation:
We have:
y = 108 / √3
d = 54 / √3

Clearly,
y = 2 × (54 / √3)
y = 2d

Final Answer:
The relation between d and y is:
y = 2d

Answer 2

Step 1: Understanding the Situation:
The value h represents the vertical height of the top of the tank above the horizontal line of sight from the window.
The horizontal distance between the building and the tank is 54 m.
Angle of elevation to the top of the tank is 45°.

Step 2: Using Sine and Cosine (Alternative Method):
Instead of directly using tangent, we first find the hypotenuse.

In right triangle BWX:
cos 45° = Adjacent / Hypotenuse
1/√2 = 54 / BW

BW = 54√2

Now use sine ratio:
sin 45° = Opposite / Hypotenuse
1/√2 = h / (54√2)

Multiply both sides:
h = (54√2) × (1/√2)
h = 54

Step 3: Conclusion:
Thus, the vertical height above the horizontal level is 54 m.

Final Answer:
h = 54 m

Answer 3

Step 1: Understanding the Concept Clearly:
The total height of the water tank (AB) is made up of two parts:
• Height below the horizontal level of the window (h)
• Height of the window above the ground (d)

So,
AB = h + d

From previous result,
h = 54 m

Step 2: Finding d Using Sine Ratio (Alternative Method):
In right triangle AWX,
Angle of elevation = 30°
Base (XW) = 54 m

First find the hypotenuse (line of sight) using cosine:
cos 30° = Adjacent / Hypotenuse
√3/2 = 54 / BW

BW = 54 × 2 / √3
= 108 / √3

Now use sine ratio to find height d:
sin 30° = Opposite / Hypotenuse
1/2 = d / BW

d = BW / 2
= (108 / √3) / 2
= 54 / √3

Rationalising:
d = (54√3) / 3
d = 18√3 m

Step 3: Total Height of Tank:
AB = h + d
= 54 + 18√3 m

Approximate value:
18√3 ≈ 18 × 1.732
≈ 31.18 m

AB ≈ 54 + 31.18
≈ 85.18 m

Final Answer:
Height of the water tank = 54 + 18√3 m
≈ 85.18 m

Answer 4

Step 1: Understanding the Geometry:
From the diagram, two right-angled triangles are formed with horizontal distance 54 m.
We use trigonometric ratios to find:
• The line-of-sight distance x (BW)
• The height of the window above ground (d)

Step 2: Finding the Line-of-Sight Distance (x):
In right triangle BWX,
Angle of elevation = 45°
Adjacent side = 54 m
Hypotenuse = x

Using secant ratio (alternative to cosine):
sec 45° = Hypotenuse / Adjacent

√2 = x / 54

x = 54√2 m

Step 3: Finding Height of the Window (d):
In the second right triangle,
Angle of elevation = 30°
Base = 54 m
Height = d

Using tangent ratio:
tan 30° = Opposite / Adjacent

1/√3 = d / 54

d = 54 / √3

Rationalising:
d = (54√3) / 3
d = 18√3 m

Step 4: Final Answer:
x = 54√2 m
Height of the window = 18√3 m