Question

In operation since 1846, the Gibb’s Hill Lighthouse stands $117$feet high on a hill $245$feet high, so its beam of light is $362$feet above sea level. A brochure states that ships $40$miles away can see the light and planes flying at $10,000$feet can see it $120$ miles away. Verify the accuracy of these statements. What assumption did the brochure make about the height of the ship?

Short answer

Hence the accuracy of the statement is true.

Step-by-step solution

Step 1. Given Information

The height of the light house is $117$feet on a $245$feet hill.

The beam of light is $362$feet above sea level.

A brochure states that ships $40$miles away can see the light and planes flying at $10,000$feet can see it $120$ miles away.

Step 2. Consider Gibb's light house

From the given information, the diagram of ship was given as follows:

Step 3. Find the distances

The total height, BD = Height of the hill(CD) + Height of the light house (BC)

$$\begin{gathered} =245+117 \\ =362 \end{gathered}$$

$ED=40miles$

$=2,11,200$feet

$FD=120$

$=6,33,600feet$

$$\begin{gathered} AG=AF-FG \\ =10,000-362 \\ =9638 \end{gathered}$$

$GB=FD$

Step 4. Find ∠DCE

Consider $∆CDE$,

$$\begin{gathered} \tan C=\frac{ED}{CD} \\ =\frac{211200}{352} \\ C={\tan }^{-1}\left(\frac{211200}{352}\right) \\ =89.90179° \end{gathered}$$

Step 5. Find ∠BED

$$\begin{gathered} \tan E=\frac{CD}{ED} \\ =\frac{362}{211200} \\ E={\tan }^{-1}\left(\frac{362}{211200}\right) \\ =0.09821° \end{gathered}$$

Step 6. Find ∠AGB

$$\begin{gathered} \tan B=\frac{AG}{GB} \\ =\frac{9638}{633600} \\ G={\tan }^{-1}\left(\frac{9638}{633600}\right) \\ =0.087149° \end{gathered}$$

Step 7. Find ∠BAG

$$\begin{gathered} \tan A=\frac{633600}{9638} \\ A={\tan }^{-1}\left(\frac{633600}{9638}\right) \\ =89.12851° \end{gathered}$$

Step 8. Assumption is true or not.

Consider the quadrilateral ABDF,

$$\begin{gathered} \angle BAG+\angle GFD+\angle FDB+\angle DBA=360° \\ 89.12851°+90°+90°+\angle DBA=360° \\ \angle DBA=90.87149° \end{gathered}$$

The angle DBA from our calculation is:

$$\begin{gathered} \angle DBA=\angle AGB+\angle GBE+\angle EBD \\ =89.90179°+0.87149°+0.09821° \\ =90.87149° \end{gathered}$$

Hence the assumption is true.