Chapter A.2, Problem 58AYU

To determine

To verify: The accuracy of this information. The brochure further states that ships 40 miles away can see the light and that 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?

Answer to Problem 58AYU

$12.25$ miles, 15 miles.

Explanation of Solution

Given:

The Gibb’s Hill Lighthouse, Southampton, Bermuda, in operation since 1846, 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 the light itself can be seen on the horizon about 26 miles distant.

[Use the facts that the radius of Earth is 3960 miles and 1 mile $=5280$ feet].

Calculation:

$1\text{ miles }=5280\text{ feet}$.

$362\text{ feet }= \frac{362}{5280}\text{ miles = }\frac{181}{2640} \text{miles}$.

By Pythagorean Theorem $c^2=a^2+b^2$.

${\left(3960+\frac{181}{2640}\right)}^2=a^2+{3960}^2$

$15682143=a^2+15681600$

$a^2=543$

$a=23.30$

No, light itself see $23.3$ miles distance.

For a plane:

$10000\text{ feet }= \frac{10000}{5280}\text{miles}$.

By Pythagorean Theorem $c^2=a^2+b^2$.

${120}^2+{3960}^2=15696000\approx 15696603={\left(3960+\frac{10000}{5280}\right)}^2$

Pythagorean Theorem satisfied.

For a ship:

By Pythagorean Theorem $c^2=a^2+b^2$.

${\left(3960+\frac{s}{5280}\right)}^2={40}^2+{3960}^2$

${\left(3960+\frac{s}{5280}\right)}^2=15683200$

$3960+\frac{s}{5280}=3960.2020$

$\frac{s}{5280}=0.2020$

$s\text{ = 1066 feet}$.

Therefore height of the ship is approximately 1066 feet.