Johnny was really excited about her trip to the Water tank. He was impressed with the way Sara had shown him how to determine the height of a tower by measuring angle and distance. It stuck in his head that Sara told him that he could measure the height of any tower this way – Statue of Liberty in New York, C.N. Tower in Toronto, Eiffel Tower in Paris or even Qutab Minar in Delhi. Also that he could look up the heights of these places and estimate the angles from different places in the cities using the map distances. He kept dreaming about this all night.
In the morning, Sara and Johnny were walking to school – of course Johnny was on his bike as usual. He told about his dream to Sara who said that they should talk about it after school. After school, Sara went home and later phoned Johnny asking if she could come over. Little did Johnny know what she had been up to. She came with a big smile on her face.
Sara: Johnny, I talked to my Internet friend Jaque DeGras in Paris and told him about your dream.
Johnny: What did he say ?
Sara: His girlfriend Sophie has done what he called “un project scolaire en geographie” last year on this subject. He introduced me to Sophie. She was excited that someone across the Atlantic was interested in her project. She measured angles using a sextant and distances from Google maps. She explained everything to me and sent me the data which I have on my laptop.
Johnny: You did all this today after school ?
Sara: Yes, it helps to have friends in places.
Johnny: So exactly what did she do in her Geography project ?
Sara: Here is the first map of the Eiffel tower area she sent me, A walking route for tourists is marked on it. She started from Hotel Duquesne Eiffel and called it spot X. Then she went over to the spot Y which was on Rue Desaex, and finally to a spot much closer to the tower she called Z. Here is her second picture of the same map.
Johnny is impressed wih Sophie’s us of the map
Johnny: I see, the map has a scale showing 200 meters. Smart girl, she just drew the lines on the map from the middle of the tower to spots X, Y and Z. From the lengths of these lines and the scale she calculated how far she was.
Sara: Yes, X was 1156 meters away, Y was 356 meters and Z was 177 meters. Here is a sketch of what she did next.
Johnny: I see that she measured two angles from spot X – one from the ground to the top of the tower (angle XAB) and the other from ground to the first floor (XAC). Then she did the same thing for the spots Y and Z. Her Table has these data.
|Spot||Distance from the tower – d||Angle for Eiffel tower||Angle for the base|
Sara and Johnny calculate height of the tower
Sara: Since the tower is vertical to the ground, we can say triangle XAB, a right angle triangle with the base XA = d and the height of the tower AB = h.
Johnny: Yes, and the same will hold for all of these right angle triangles. Then height/base = tan (XAB) and h = d tan (XAB). This will give us all the heights. Let us put them in a Table.
|Spot||d in meters||Eiffel tower||First floor|
|X||1156 meters||15.7⁰||325 meters||2.9⁰||59 meters|
|Y||356 meters||42.3⁰||324 meters||9.3⁰||58 meters|
|Z||177 meters||67.7⁰||324 meters||23.6⁰||58 meters|
Sara: Johnny you did it. Here is your dream come true. Also, the height given for the Eiffel tower is 324 meters. So you have the right answer.
Johnny: Thanks Sara. We should thank Sophie too.
Sara: I already did. She was also very happy to bring your dream into reality. I think, one weekend we should go see the Statue of Liberty.
Johnny: I would love that.
Billu likes to fly his kite from a playground near his home. Today, the wind is blowing and his kite is flying well such that the angle between the ground and the string is 30⁰. The wind is in the direction of his school which is 1 kilometer away. He has only 1.5 kilometer long string and is wondering if it would be long enough for the kite to reach over the school.
Here is the picture. His starting point is A, and the school is at point B. A vertical line can be drawn from B to meet the string at C. In this right angle triangle, angle BAC = 30⁰ (given).
AB/AC = cos 30⁰ = 0.866.
Since AB = 1 kilometer, 1/AC = 0.866 or AC = 1/0.866 = 1.1547 kilometers.
Therefore, he has more than enough string (1.5 kilometers) for he wants to do.