(A). The angles of depression of two ships from the top of a lighthouse are 60° and 45° towards the east. If the ships are 300 meter apart, the height of the lighthouse is $150(3+\sqrt{3})$

(B). If the surface area of a cube is $726 m^2$, then its volume shall be $1331 m^3$

(C) If the ratio of diameters of two spheres is 3 : 5, then the ratio of their surface area shall be 9 : 25

Determine as to which of the statements given above are correct.

(A), (B) & (C)

The correct answer is Option (2) → (A), (B) & (C)

Let us check each statement one by one.

Statement (A)

Let the height of the lighthouse = h

• Angle of depression = $60^\circ$:
  $\tan 60^\circ = \frac{h}{d_1} \Rightarrow d_1 = \frac{h}{\sqrt{3}}$
• Angle of depression = $45^\circ$:
  $\tan 45^\circ = \frac{h}{d_2} \Rightarrow d_2 = h$

Distance between ships =

$d_2 - d_1 = h - \frac{h}{\sqrt{3}} = h\left(1 - \frac{1}{\sqrt{3}}\right) = 300$

Solving,

$h = \frac{300\sqrt{3}}{\sqrt{3}-1} = 150(3+\sqrt{3})$

Statement (A) is correct

Statement (B)

Surface area of cube = $6a^2 = 726$

$a^2 = 121 \Rightarrow a = 11$

Volume = $a^3 = 11^3 = 1331\ \text{m}^3$

Statement (B) is correct

Statement (C)

Surface area of a sphere ∝ $\text{(diameter)}^2$

Given ratio of diameters = 3 : 5

Surface area ratio $=\text{Surface area ratio} = 3^2 : 5^2 = 9 : 25$

Statement (C) is correct

Correct Answer: (A), (B) & (C)