Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# How high is that cloud? How do we measure something so far and so high up like a mountain top, or a cloud? Trigonometry comes to mind. But we can't physically measure out long distance, what should we do?![[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.33.25.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.33.25.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.33.25.excalidraw.dark.svg|dark exported image]]%% Let us develop a method using trigonometry. Maybe we call it the stepping forward/backward method. ## Stepping forward/backward method. Consider the following right triangle: ![[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.30.51.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.30.51.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.30.51.excalidraw.dark.svg|dark exported image]]%% And suppose we are interested in finding out $H$ and/or $B$. But these are impractically large distances to measure out physically. However, we are able to find the angle $\alpha$ made by this right triangle, opposing the side $H$. And if we are to step back a known distance $x$, we would form a new right triangle with angle $\beta$ opposing the side $H$: ![[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.40.11.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.40.11.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.40.11.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] Solve for $H$ in terms of angles $\alpha$, $\beta$, and the distance $x$. For this to be useful, $H$ should not depend on $B$. Hint: To troubleshoot, your answer should involve $\tan(\alpha)$ and $\tan(\beta)$... ![[---images/---assets/---icons/question-icon.svg]] Solve for $B$ in terms of the angles $\alpha, \beta$, and the distance $x$. For this to be useful, $B$ should not depend on $H$. Congratulations! We know have a method to find the distance and height of a far away object, by knowing the initial **angle of elevation** $\alpha$, and the **final angle of elevation** $\beta$, by stepping back a known distance of $x$ ! ## Finding the angle of elevation. ![[---images/---assets/---icons/question-icon.svg]] Using a piece of string, some weight, and a cardboard with markings, can you imagine how one could measure the angle of elevation? ![[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.55.08.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.55.08.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/how-high-is-that-cloud 2023-08-21 14.55.08.excalidraw.dark.svg|dark exported image]]%% The ancients build many devices that measure the angular separation between two objects, such as the sextant, octant, and the quadrant. These days, however, we can leverage a powerful technology in our pocket: **Our mobile phone** with camera. Here I made a very crude digital sextant to measure the angle of elevation : [digital sextant](https://bonsoon.net/dev/sextant/) However, the sensors I'm using in your phone (2023) can only detect up to $0.1^{\circ}$ change in angle, so it can be quite limited. ## Let's find some heights! ![[---images/---assets/---icons/question-icon.svg]] Use the digital sextant (or build one with strings and markings), estimate the height of the ceiling in our class room. Notice, you would need to account for the height of your eye level, as well as how far you step back. A standard sheet of paper is $8.5\text{ inches}\times11\text{ inches}$ for your reference, if you do not have a ruler. **There is a bonus exercise if you estimate the height of Royce Hall: [[1 teaching/summer program 2023/puzzles-and-problems/how-tall-is-royce-hall|how-tall-is-royce-hall]]** This method, though mathematically correct, is very sensitive on the angle you measured, as well as the distance $x$ (usually larger the $x$ the better). And that you will need to keep everything steady and at the same constant height. Work together to see if you make this method as good as possible by reducing some of those factors. Comment on the errors and improvements that can be made with this method.