cintinative Posted March 1, 2022 Share Posted March 1, 2022 We are in Dolciani Alg II. I am working ahead of my son. I really stink at drawing pictures to get the problem set up for a word problem. For example, this one: "A monument consists of a flagpole 15 m tall standing on a mound in the shape of a cone with vertex angle 140 degrees. How long a shadow does the pole cast on the cone when the angle of elevation of the sun is 62 degrees?" I literally wasn't sure where to draw the shadow or where the elevation of the sun angle should be drawn. I looked in the solution manual and their drawing didn't make sense. I took to Google and found a drawing that did make sense. But I want to be able to draw this from the description and I just seem to lack this skill. I am not strong in visual-spatial stuff--that's always been a weakness. That might be part of my issue? Where can I get help? Other than just looking in the solution manual to see how they drew it? Is there a way to practice this? Quote Link to comment Share on other sites More sharing options...
Alice Lamb Posted March 1, 2022 Share Posted March 1, 2022 Am I missing something, or don't you need some information about the height or diameter of the base of the mound? Is this in a chapter on basic trig functions in a triangle? Quote Link to comment Share on other sites More sharing options...
cintinative Posted March 1, 2022 Author Share Posted March 1, 2022 37 minutes ago, Alice Lamb said: Am I missing something, or don't you need some information about the height or diameter of the base of the mound? Is this in a chapter on basic trig functions in a triangle? It's in a chapter on law of sine and cosine. I can solve it once I can draw it. The trouble is drawing it. 😃 Quote Link to comment Share on other sites More sharing options...
daijobu Posted March 1, 2022 Share Posted March 1, 2022 Boy, AoPS is way easier than this. The most complicated flag pole problem you'll see in AoPS is something like 2 flagpoles of different sizes and you know their shadow lengths. Or maybe a couple of flag poles and you need to figure out how far apart they are and it could be cloudy or sunny, who knows. I don't even know what the "angle of elevation" of the sun even means. I'm guessing since it's a mound shape, then the cone is inverted so that the point of the cone coincides with the base of the flagpole? But yeah, how big is this mound? Typically I picture a flagpole with a mound of maybe a few inches just to stabilize it, while the flagpole is 10 feet tall. If it were close to noon maybe I could see a really short shadow. 1 Quote Link to comment Share on other sites More sharing options...
cintinative Posted March 1, 2022 Author Share Posted March 1, 2022 3 hours ago, daijobu said: Boy, AoPS is way easier than this. The most complicated flag pole problem you'll see in AoPS is something like 2 flagpoles of different sizes and you know their shadow lengths. Or maybe a couple of flag poles and you need to figure out how far apart they are and it could be cloudy or sunny, who knows. I don't even know what the "angle of elevation" of the sun even means. I'm guessing since it's a mound shape, then the cone is inverted so that the point of the cone coincides with the base of the flagpole? But yeah, how big is this mound? Typically I picture a flagpole with a mound of maybe a few inches just to stabilize it, while the flagpole is 10 feet tall. If it were close to noon maybe I could see a really short shadow. You have just made me feel so much better. This is a level A word problem (so, the easiest). I have really struggled with the angle of elevation/angle of depression problems. They have had some where it was an airplane flying in and they were sighting two parts of a runway at certain degrees measurement, and I would try to draw it and would draw it wrong every single time. Sigh. This problem was solved (and "sketched") here, if anyone is curious. I had no idea how they knew where the shadow falls, and for the most part, the fact it is a cone means very little other than that the sides are slanted. https://www.freemathhelp.com/forum/threads/trig-story-problems-please-help-tonight.63221/ Quote Link to comment Share on other sites More sharing options...
daijobu Posted March 1, 2022 Share Posted March 1, 2022 Ah, ASCII art! You don't see that very much anymore. So it looks like this so-called "mound" is actually a hill? Because it looks like it's actually taller than the flagpole itself, that is, . Remind me not to use Dolciani. 1 Quote Link to comment Share on other sites More sharing options...
cintinative Posted March 1, 2022 Author Share Posted March 1, 2022 21 minutes ago, daijobu said: Ah, ASCII art! You don't see that very much anymore. So it looks like this so-called "mound" is actually a hill? Because it looks like it's actually taller than the flagpole itself, that is, . Remind me not to use Dolciani. Yes, BE represents the hill/mound height and is taller than the flagpole. I guess this is an imprecise language thing because to me a hill and a mound could be the same thing, or they could be terribly different. I really do like Dolciani overall, but my struggle to get these problems set up properly because I can't draw them has been pretty humbling. Here's that runway problem I couldn't draw: "A pilot approaching a 10,000 foot runway finds that the angles of depression of the ends of the runway are 12 degrees and 15 degrees. How far is the plane from the nearer end of the runway?" Quote Link to comment Share on other sites More sharing options...
mom31257 Posted March 2, 2022 Share Posted March 2, 2022 (edited) NM I saw you found it Edited March 2, 2022 by mom31257 Quote Link to comment Share on other sites More sharing options...
mom31257 Posted March 2, 2022 Share Posted March 2, 2022 (edited) Here is the other problem. Edited March 2, 2022 by mom31257 Quote Link to comment Share on other sites More sharing options...
cintinative Posted March 2, 2022 Author Share Posted March 2, 2022 So no one has any advice on how I can build my skill in this? Do I just watch other people set up the problems? I feel like I am cheating to look in the solution manual to know how to set them up. Quote Link to comment Share on other sites More sharing options...
mom31257 Posted March 2, 2022 Share Posted March 2, 2022 20 minutes ago, cintinative said: So no one has any advice on how I can build my skill in this? Do I just watch other people set up the problems? I feel like I am cheating to look in the solution manual to know how to set them up. I try to draw them exactly as it says. Is it the angles of depression and elevation are the problem? Or do you struggle with all drawings? Quote Link to comment Share on other sites More sharing options...
cintinative Posted March 2, 2022 Author Share Posted March 2, 2022 24 minutes ago, mom31257 said: I try to draw them exactly as it says. Is it the angles of depression and elevation are the problem? Or do you struggle with all drawings? I guess the angles of depression and elevation, if you include the angle of the sun and the way a shadow falls in that. 😃 Maybe I should just watch some videos on that? Once I see it drawn, it makes sense, but I can't figure out how to draw it from the word problem sometimes. Quote Link to comment Share on other sites More sharing options...
mom31257 Posted March 2, 2022 Share Posted March 2, 2022 (edited) I've just taught these in my geometry course. The important thing is to draw a horizontal from the object parallel with the ground. Angles of depression are always looking down to the ground and are above the line of sight. Angles of elevation are looking up and below the line of sight. The parallels create alternate interior angles, making it easy to figure out angles in the triangles created. https://www.mathsisfun.com/definitions/angle-of-depression.html https://www.onlinemathlearning.com/angle-of-elevation.html Edited March 2, 2022 by mom31257 1 Quote Link to comment Share on other sites More sharing options...
Alice Lamb Posted March 2, 2022 Share Posted March 2, 2022 I'm still not clear on how the height of the mound/hill is really irrelevant. If the height of the mound is quite small, say 1 foot, with the same angle at the top of the cone, the base of the mound would only have a relatively small radius and the shadow would be mostly on the flat ground beyond the mound. Then, wouldn't triangle AEC be used to calculate the shadow length AC? 1 Quote Link to comment Share on other sites More sharing options...
mom31257 Posted March 3, 2022 Share Posted March 3, 2022 (edited) 6 hours ago, Alice Lamb said: I'm still not clear on how the height of the mound/hill is really irrelevant. If the height of the mound is quite small, say 1 foot, with the same angle at the top of the cone, the base of the mound would only have a relatively small radius and the shadow would be mostly on the flat ground beyond the mound. Then, wouldn't triangle AEC be used to calculate the shadow length AC? I take take it to mean that the shadow is only on the mound. Edited March 3, 2022 by mom31257 Quote Link to comment Share on other sites More sharing options...
daijobu Posted March 3, 2022 Share Posted March 3, 2022 12 hours ago, cintinative said: So no one has any advice on how I can build my skill in this? Do I just watch other people set up the problems? I feel like I am cheating to look in the solution manual to know how to set them up. I don't think it's cheating. I would look at the diagram in the solution manual and try to understand why it is correct. Then let a day or two elapse and attempt the same problem again. See if you can improve translating words into drawings. 1 Quote Link to comment Share on other sites More sharing options...
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