Light Houses Galore

north shore lighthouses
Help with a calculus question?

A lighthouse is positioned off the coast so that the shortest distance between the lighthouse (L), and a hotel on the straight beach (H) is .4 miles. The lighthouse lens spin around, counterclockwise, at a rate of 5 revolutions per minute. How fast is the beam moving across the shore when the beam's distance (north) of the hotel is (1/3) miles?

A picture is the best way to start the process. As you can see, a right triangle is formed with the distance from the lighthouse to the hotel, y, being a constant .4 miles. The distance from the beam to the hotel, x, varies. So does the distance, r, from the lighthouse to the shore.

Clearly at any time, r^2 = x^2 + y^2. We can differentiate this with respect to time giving 2r(dr/dt) = 2x(dx/dt) + 2y(dy/dt). Solving for dx/dt will give us the desired result.

dx/dt = [r(dr/dt) - y(dy/dt)] / x

Since the lighthouse is always 0.4 miles from the shore, it is clear that dy/dt is zero. We can determine r and x from the Pythagorean Theorem at the instant where the beam is 1/3 miles north of the hotel. So, x = 1/3 miles at that point, y = 0.4 miles and r = 0.5207 miles at that instant in time. That leaves the sticky problem of determining dr/dt.

Now, dΘ/dt = 5(2π) = 10π radians per minute. Also, we can establish that cos(Θ) = 0.4/r from our diagram. Solving for r = 0.4/cos(Θ). From this relationship, we can compute dr/dΘ = 0.4tan(Θ)sec(Θ). Finally, we can compute dr/dt = (dr/dΘ)(dΘ/dt) = 0.4tan(Θ)sec(Θ)(10π) miles/min. At the instant in question cos(Θ) = 0.4/.5207 = .7682, so Θ = 39.8 degrees.

We now have everything we need to compute the answer.

dx/dt = [r(dr/dt) - y(dy/dt)] / x

Plugging in gives:

dx/dt = (.5207)( 0.4tan(39.8)sec(39.8)(10π) / (1/3) miles/min

dx/dt = 21.28 miles / min

Hopefully, my analysis and calculations are correct. Good Luck.

Five Historical Australia Attractions

Australia is undoubtedly the perfect holiday destination. The rich cultural history of the country is known all around the world and there are a number of monuments and places that represent the richness of the culture. Here is a sneak peek at five of the Australia attractions. Add them up in your itinerary while you plan tours to Australia.

Barrenjoey lighthouse in Sydney was built in the year 1881 and as the name suggests, it was used to make the sailors aware of the shore. The lighthouse is made of sandstone and watching the sea waves banging against the shore is no less than scenery. Other methods were used to warn the sailors in the past but due to the nature of the sea, the shore needed a lighthouse. The lighthouse still is one of the best Australia attractions and you too should pay a visit to this beautiful place that offers great scenic views.

Another landmark in the history of the country is the Chinese Garden of Friendship. It is one of the most loved destinations for people on tours to Australia. The city Guangdong city, which is known to be the sister city of Sydney, gifted this garden to Australia on its bicentennial in the year 1987. Known for its peaceful and serene environment, this garden will find you awestruck while you witness its beauty. Experience the Chinese environment while you roam around in one of the best Australia attractions.

When it comes to monuments in Australia, there may be many of them but one of the most important ones is the Parliament House Sydney. The building of the house dates back to the year 1816 but some more additional buildings were added in the later years. While on Australia tours, travelers and general public can visit the gallery of the house.

History is not always related to politics and religious events. There are some places that are different than others and have no direct connection with the politics of the country. One of the most visited Australia attractions is the Quarantine station. A station tour becomes a must for the travelers if they want to add an exciting horror element to their tours to Australia. The travelers who entered the port of Australia in the 19th and 20th century were quarantined here and the equipments and stone cottages are still present there. It is said that many travelers stayed here and died.

If you want to witness an amazing example of engineering in Australia for tour itineraries, pay a visit to the Sydney Harbor Bridge. The Sydney Harbour Bridge is a steel arch bridge across Sydney Harbour that carries rail, vehicular and pedestrian traffic between the Sydney central business district and the North Shore. The dramatic view of the bridge, the harbour, and the nearby Sydney Opera House is an iconic image of both Sydney and Australia. The bridge is locally nicknamed "The Coathanger" because of its arch-based design. This bridge was completed in the year 1932 and can easily manage traffic that passes by in a total of eight lanes. This is also world’s widest long span bridge and become a symbol of recognition for citizens of this amazing country.

About the Author

Raj Aryan is a content writer. Presently working with a Tour Package company. Find India vacation packages and online at Cheap flights tickets to India, providing the best selection and availability of cheap tickets.

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