Question

A sailor, standing on the deck of a ship, just sees the light beam from a lighthouse on the shore. If the height of the sailor's eye and the light beam at the lighthouse, above the sea level, are \(9\) m and \(25\) m respectively, what is the approximate distance between the sailor and the lighthouse?

• A. \(27\) km
• B. \(31\) km
• C. \(35\) km
• D. \(39\) km

Hint:

For intervisibility between two elevated points, \[ \boxed{ D=3.86(\sqrt{h_1}+\sqrt{h_2}) } \] where \(h_1\) and \(h_2\) are in metres and \(D\) is in kilometres.

Answer 1

The Correct Option is B

Concept: The distance to the horizon from an object of height \(h\) (in metres) is \[ \boxed{d=3.86\sqrt{h}} \] where \(d\) is in kilometres. For two elevated points just visible to each other, \[ \boxed{ D=3.86\left(\sqrt{h_1}+\sqrt{h_2}\right) } \]

Step 1: Write the given data. \[ h_1=9\text{ m},\qquad h_2=25\text{ m} \]

Step 2: Calculate the distance. \[ D = 3.86(\sqrt{9}+\sqrt{25}) \] \[ = 3.86(3+5) \] \[ = 3.86\times8 = 30.88\text{ km} \] \[ \boxed{D\approx31\text{ km}} \] Therefore, the correct option is \[ \boxed{(B)\;31\text{ km}.} \]