Question

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37. From the top a light house, angles of depression of two ships are 45 and 60°. The ships are on opposite
sides of the light house and in line with its foot. If the distance between the ships is 400 m, find the
height of the light house.
Answer the following question.​

Answer 1

❥${\red{\underline{\underline{\huge{\mathtt{Question:-}}}}}}$

From the top a light house, angles of depression of two ships are 45° and 60°. The ships are on opposite sides of the lighthouse and in line with its foot.If the distance between the ships is 400 m, find the height of the lighthouse.

❥${\red{\underline{\underline{\huge{\mathtt{Solution:-}}}}}}$

From the top of the lighthouse,

• Angle of depression of 1st ship(angle EAB) = 45°.

• Angle of depression of 2nd ship(angle FAC) = 60°.

• The distance between two ships(BC) = 400 m.

Consider,

The distance from the 1st ship to the lighthouse (BD)= x m.

So, the distance from the 2nd ship to the lighthouse (DC)= (400-x) m.

• angle EAB = angle ABD = 45° [ Alternate angles]
• angle FAC = angle ACD= 60° [ Alternate angles]

★ In case of ∆ ADB,

$\frac{AD}{BD}$= tan45°

→$\frac{AD}{x}$= 1

→ AD = x ............(i)

★In case of ∆ ADC,

$\frac{AD}{DC}$=tan60°

→$\frac{x}{(400-x)}$= √3

→x =400√3 - x√3

→x +x√3 = 400√3

→x(1+√3)= 400√3

→x(1+1.732)= 400×1.732

→2.732 x =692.8

→x = 253.59 ( Approx)

✞Putting x = 253.59 in (i) no. equation✞

AD = 253.59 (Approx)

❥${\red{\underline{\underline{\huge{\mathtt{Answer:-}}}}}}$

Hence, the height of the lighthouse is 253.59 m (Approx).

Answer 2

$\huge \mathfrak{Answer:-}$$\large\underline\mathrm{Given:-}$

• angles of depression of two ships are 45 and 60°.
• the distance between the ships is 400 m.

$\large\underline\mathrm{To \: find}$

• The height of the light house.

$\large\underline\mathrm{Solution}$

• The distance from the 1st ship the lighthouse (BD) = x m.

$\large\underline\mathrm{so,}$

$\large\underline\mathrm{The \: distance \: from \: the \: 2nd \: ship \: to \: the \: lighthouse \: (DC) \: = \: (400 \: - \: x)m.}$

$\implies$EAB = ABD = 45°. [Alternate angle]

$\implies$FAC = ACD = 60°. [Alternate angle]

$\large\underline\mathrm{In \: ∆ADB,}$

$\implies$AD/BC = tan 45°

$\implies$ AD/x = 1

$\implies$ AD = x_______(1)

$\large\underline\mathrm{In \: ∆ADC,}$

$\implies$ x/(400 - x) = √3

$\implies$ x = 400√3 - x√3

$\implies$ x + x√3 = 400√3

$\implies$ x(1 + √3) = 400√3

$\implies$ x(1 + 1.732) = 400 × 1.732

$\implies$ 2 732x = 692.8 = 253.58

$\implies$ x = 253.59

$\implies$ AD = 253.59

$\large\underline\mathrm{hence,}$

$\large\underline\mathrm{the \: height \: of \: the \: lighthouse \: is \: 253.59m.}$

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$