Chapter 8.1, Problem 83AYU

The Gibb’s Hill Lighthouse, Southampton, Bermuda 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?

To determine

To verify: The accuracy of the given statements. What assumption did the brochure make about the height of the ship?

Answer to Problem 83AYU

The statement about the plane is correct.

Explanation of Solution

Given:

The Gibb’s Hill Lighthouse, Southampton, Bermuda 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.

Formula used:

Pythagoras theorem.

Calculation:

Let $\theta$ be the central angle formed by the top of the light house, the centre of the earth and the point $P$ on the Earth’s surface where the line of sight from the top of the light house is tangent to the Earth.

$d1=\sqrt{{\left( 3960+\frac{362}{5280}\right)}^2-{3960}^2}$

$d1=23.3 \text{miles}$

So the first statement about the ship is false.

If we take height of the ship into account, let the height be $x$.

$d2=40-23.3=16.7$

${(3960+x)}^2= \left({16.7}^2+{3960}^2\right)$

$x=185.9\text{ ft}$

If $x=10,000\text{ ft}$ as height of plane.

$d2=\sqrt{({\left(3960+\frac{10000}{5280}\right)}^2-{3960}^2)}$

$d2=122.48 \text{miles}$

Hence, The statement about the plane is correct.