Evening stroll on a summer day
On most days, Sara and Johnny used to walk together just to school. On some occasions, they also loved going for evening strolls. This was one of those days. It had been a hot day but the heat was quite bearable after the sun had already set. In fact, frequent breeze made it very comfortable. It was that the time of the year when the trees were green and there were flowers were colorful. Sara suggested that they should go for a stroll to the city water tank tower. The water tower was in the center of the town and many people went there and enjoyed just sitting on benches in the park next to it. It was not that far from where Johnny and Sara lived. Even then Johnny took his new bike with him. He loved the bike so much and took it everywhere. Surprisingly, Sara had a small box in her hand. Johnny wondered what it was and why she was carrying it.
Johnny: Sara, what’s with the box?
Sara: Actually, the box belongs to my dad. My grandfather gave it to him as a present when my dad was a teenager. My dad cherishes it but he let me borrow it for today.
Johnny: What’s in it? The suspense is killing me.
Sextant – an instrument for measuring angles
Sara: The box contains a Sextant – an instrument that civil engineers use for measuring angles. My grandfather thought that my dad could become one. Instead, my dad went for software engineering. It was a good thing because that’s where he met my mom.
Johnny: What are we going to do with it?
Sara: We’ll use it while watching the water tank tower.
Johnny: How high is the water tower?
Sara: That’s one of the things we are going to find out.
Johnny: Even if I climbed the tower, how will I measure its height? I don’t even have a measuring tape.
Sara: That’s why I brought my dad’s sextant. Let’s go to the base of the tower and then walk away from it.
Johnny: How far?
Measuring the height of the tower
Sara: 150 meters, you can measure it with your fancy bike.
They both kept walking until Johnny’s bike said that they had moved 150 meters West from the tower. Sara measured two angles from there. One was the angle from the base of the tower to the top of the water tank. She called this angle X, and it was 35°. The second angle she measured was Y which was from the bottom of the water tank to the base of the tower. This angle was 30°. Sara wrote these down and they continued enjoying the evening. Johnny was getting restless. How could she measure the height of the tower with this, he wondered.
They returned to Sara’s home where they had something to drink, and then Sara drew a picture (Fig. 3.1). By now, Johnny had developed enough confidence in Trig to follow the meaning of this picture. Soon he realized what Sara was up to.
Johnny: Are we going to use the angles to figure out the height of the tower? It should be easy. For angle X, the height of the top of the tower divided by the distance to the base will be tan X.
Sara: Yes, angle X = 35° and my calculator shows that tan 35° is 0.7. So that means the ratio DC/AB (height to the base) = 0.7.
Johnny: I measured with my bike that the length of the base (AB) was 150 meters. That means for the height DC, DC/150 = 0.7 or DC = 105 meters. Wow, the tower is 105 meter high.
Sara: Yes, and you figured it out without actually climbing it.
Capacity of the water tank
Johnny: Without the tank, the angle is only 30° and tan 30° = 0.577. So without the tank, the tower would be only 86.55 meters (150 x 0.577) high. So the tank adds a height of 18.70 meters to the tower. That’s a big tank. It can hold a lot of water.
Sara: The tank looks like a sphere. Its radius would be half its height, say about 9.35 meters. I checked online that the volume of a sphere equals 4πr3/3 where r is its radius. This volume would come out to be about 3200 cubic meters or 3.2 million liters.
Johnny: That’s a lot of water. I can drink only about 3 liters of water in one day. It would take me a million days to drink it all. Here is the website of the city. It lists the water tank to be 105 meters high with the tank. That’s what we calculated. It says that the tank can hold 2.9 million liters of water.
Sara: We overestimated the tank capacity because we measured only the outer dimensions of the tank and did not consider the thickness of the wall of the tank.
Johnny: So that means we can 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.
Sara: Yes, if we can go there with my dad’s sextant and your bike. However, for sure we can look up the heights of these places and estimate the angles from different places in the cities using the map distances between.
Johnny thought this idea of finding heights could be applied to anything – towers, cliffs, skyscrapers, mountains and so on. He was both excited and impressed with the new powers he had developed. He found out at home that the book gave ample support to this thought. He told his dad about it, especially since his dad’s company constructed large buildings. There were a lot of questions in the book using these ideas. He did many more questions of this type in the next couple of days. Yes, as a Trig whiz he was ahead of the lessons being taught in the class. Many classmates came to Johnny asking for help with their Trig homework. This was a new experience for him.
Challenge: The city of Crapstone does not have an airport and wants to design one on the only land that is available to it on North of the city. A runway cannot be designed in the East-West direction due to many geographical problems and hence a North-South runway has to be considered. The city has several tall buildings 3 kilometers away from the southern end of the possible runway – the tallest being 100 meters high. North of the airport is out of question due to a mountain. Would it be a problem considering that the angle of descent during landing is minus 3° and the angle of ascent is being planned to be 8°?
Solution: Draw a right angle triangle starting from the southern end of the run way (A) to the 100 meters high building (BC) at risk at 3 km (3000 meters) (Fig. 10.4).
Given: AB = 3 km = 3000 meters.
Question: What is the value of the angle of CAB (x)?
tan x = BC/ AB = 100/3000 = 0.0333. Arctan (.03333) = 1.91° which is less than 3° or 8°. The height of the plane at B will be 3000 x tan (3°) = 157.2 meters during descent and
3000 tan (8°) = 421.6 meters during ascent.
The planes will not crash on the building during ascent or descent.