Statue of Liberty
Statue of Liberty is a copper statue, gifted from the people of France to the people of the United States. It was designed by French sculptor Frédéric Auguste Bartholdi and its metal framework was built by Gustave Eiffel. The statue was dedicated on October 28, 1886. Millions of tourists go to see this spectacular pride of franco-american friendship every year. Cruises in the Hudson river are available for sightseeing on a regular basis.
On this long weekend, Johnny asked his mom if she could take him and his girlfriend Sara to see the Statue of Liberty. She happily agreed. Sara also took permission from her Nana for the same. They decided to go for the cruise. Johnny enquired and discovered that he could take his bike on the boat. He loved the idea. Sara also took her dad’s sextant along.
The boat stopped momentarily
They were in the boat which has stopped momentarily to give people a view of the monument. The boat was facing the statue. Johnny and Sara went to the back of the lower deck of the boat where Sara measured that the angle from the ground to the top of the statue was 25⁰. Immediately, they walked to the front of the still boat. Johnny measured on his bike that they had walked 80 meters. Now Sara measured the angle to be 37.8⁰.
When they came home, Johnny drew this picture in which the first angle (XAB) had been measured at point X in the back of the boat and the second (YAB) at point Y in the front. X and Y were 80 meters apart.
Sara: Let us say that the distance AY was d meters. Then AX would be d + 80 meters. If the height of the statue was h meters, then
h/d = tan 37.8⁰ and h/(d+80) = tan 25⁰.
Johnny: I checked online that tan 37.8⁰ = 0.775 and tan 25⁰ = 0.466.
That means h/d = 0.775 and h/(d+80) = 0.466 or
d/h = 1.29 ….equation 1, and
(d+80)/h = 2.15… equation 2.
Subtracting equation 1 from equation 2 gives
80/h = 2.15 -1.29 = 0.86 or 80 = 0.86 h or 93 = h.
Sara: Hey, you are now a Trig expert. The brochure they gave us at the cruise said that Statue of Liberty is 93 meters high from the ground level.
Johnny loved his new found skill of using Trig to measure heights.
Johnny and Sara’s Mountain Challenge
Johnny and Sara went onto a mountain with a winding road. They were at the spot A shown in the picture and wanted to hike to spot B.
Johnny: I went on my bike on the winding road from A to B the other day. It was 5 kilometers by road and the average inclination of the road was 4⁰.
Sara: So that is a height of 5000 sin (4⁰) = 5000 x 0.07 = 350 meters.
Sara took out the small range finder from her purse and measured that the distance between A to B it was 500 meters.
Johnny: So that is hike at an angle of arcsin (350/500) which is arcsin (0.7) which is a slant of 44⁰.
Sara: That’s too steep to hike. We are better of walking 5 kilometer on the road.
Johnny: We could walk to the left rather than straight up and then turn right half way up to reach to our destination.
Sara was convinced. They ended up going on the path shown and reached point B.
Johnny: My fitness measure shows that we walked 800 meters.
Sara: So that would mean that we walked at an angle of arcsin (350/800) or arcsin (0.4375). My calculator shows that it would be an angle of 26⁰ which was much better to manage than the hike at an angle of 44⁰.