(A). The angles of depression of two ships from the top of a lighthouse are $60^\circ$ and $45^\circ$ 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 $726 \text{ m}^2$, then its volume shall be $1331 \text{ m}^3$

(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 :

Statement (A): Two ships are towards the east with angles of depression 60° and 45° from a lighthouse. The ships are 300 m apart, and the claimed height is $150(3+\sqrt{3})$ m.

Let $h$ = height of lighthouse.

The angle of depression equals the angle of elevation from the ship to the top.

For the ship with 60° depression:

$\tan(60°) = \frac{h}{d_1}$

$\sqrt{3} = \frac{h}{d_1}$

$d_1 = \frac{h}{\sqrt{3}}$

For the ship with 45° depression:

$\tan(45°) = \frac{h}{d_2}$

$1 = \frac{h}{d_2}$

$d_2 = h$

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Since both ships are towards the east, the distance between ships:

$d_2 - d_1 = 300$

$h - \frac{h}{\sqrt{3}} = 300$

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

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

$h = \frac{300\sqrt{3}}{\sqrt{3} - 1}$

Rationalizing:

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

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

$h = \frac{300\sqrt{3}(\sqrt{3} + 1)}{2}$

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

$h = 150(3 + \sqrt{3})$

Statement (A) is correct.

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Statement (B): A cube has surface area 726 m² and claimed volume 1331 m³.

Surface area of cube $= 6a^2$

$6a^2 = 726$

$a^2 = 121$

$a = 11$ m

Volume of cube $= a^3$

Volume $= 11^3$

Volume $= 1331$ m³

Statement (B) is correct.

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Statement (C): Two spheres have diameter ratio 3:5, and claimed surface area ratio 9:25.

If $d_1:d_2 = 3:5$, then $r_1:r_2 = 3:5$

Surface area of sphere $= 4\pi r^2$

Ratio of surface areas:

$\frac{SA_1}{SA_2} = \frac{4\pi r_1^2}{4\pi r_2^2}$

$\frac{SA_1}{SA_2} = \frac{r_1^2}{r_2^2}$

$\frac{SA_1}{SA_2} = \frac{3^2}{5^2}$

$\frac{SA_1}{SA_2} = \frac{9}{25}$

Statement (C) is correct.

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All three statements (A), (B), and (C) are correct.