I got a few things done this week on KA, but overall, my effort was pretty lousy. I made it through the unit test for Analytic Geometry and then got through two exercises in the final unit, Solid Geometry. Considering my goal was to get through all four exercises AND the unit test however, the week was a fail. I probably only studied for 3-4 hours at the most which was part of the problem and disappointing to say the least. There were a few highlights that I’ll get into below, and I’m glad I at least made some progress, but given how close I am to finishing this whole thing off, I feel like I should be pushing myself a bit harder to get it all done. I know I’ll get through it all, but I’m worried I’ll be an old man by the time I do. 👴🏼

Here are three example questions from Analytic Geometry’s unit test:

Unit Test – Analytic Geometry

Question 1

I ended up getting this question wrong because I didn’t do Pythag’s Theorem properly going from X to where the point “Eric” was. I can’t remember exactly what I did wrong, but after I did the algebra, the value I got under the radical was less than (89)^1/2 so I went with (89)^1/2= ~9.4 which was wrong. This was one of the only few careless mistakes I made during the test which I was happy about but I was still annoyed. 😠

Question 2

This was a pretty easy question to answer. I was glad that I instantly knew that to solve it I’d need to determine the slopes of both lines and if they were equal, the lines would be parallel, if they were negative reciprocals of each other, they’d be perpendicular, and if they were anything else, they’d be neither. As you can see, I quickly solved the slope for each, realized the slopes were negative reciprocals of each other, and determined they were perpendicular. Boom. 🧨

Question 3

As you can see, I used Desmos to solve this question which I probably wasn’t supposed to do so I cheated in that sense. BUT, I was pumped that I actually remembered what the equation for a circle was (x² + y² = r) and that I knew how to shift the circle and expand its radius which you can see in the equation I wrote on Desmos.

I finished the unit test on Thursday and then got started on the next unit, Solid Geometry, on Friday. As I mentioned, there were four exercises I needed to get through, three of which came from a section titled Cavalieri’s Principle and Dissection Methods that must not have been in the unit when I first did it five years ago. The section had three articles, four videos, and three exercises, all of which I hadn’t started/done. I read the first two articles and was able to get the gist of Cavalieri’s principle which can sort of be summed up with this:

This was from the second of the two articles. The first article was about the principle in 2D. The point is, if two shapes have the same height and “cross-sectional area” (kind of like the width at every slice up and down the shape, but not exactly), then they’ll have the same volume. (The image of the poker chips is a good metaphor sum up the principle.) After going through those articles, I didn’t bother reading the other article or watching the videos and jumped straight into the exercises.

Exercise 1 – Apply Cavalieri’s Principle

Question 4

This question and the next were super obvious to answer, so even though I didn’t know much about Cavalieri’s Principle, I got them both correct just because they were stupidly obvious. I don’t know what else I should say about these questions since I don’t have a strong grasp on Cavalieri’s principle, but I’ll say that anyone with a general understanding of shapes/volume would be able to intuit the solution to these questions.

Question 5

Exercise 2 – Use Related Volumes

Question 6

I thought of the poker chips when solving this question. It seemed obvious to me that just because the second cylinder was slanted, the volume wouldn’t be any different. Turns out, I’m a complete genius and got it correct. 🤓 

Question 7

This question was a little trickier to answer, and I wasn’t sure if I’d be correct when submitting my answer, but it seemed to me that since the height of the larger pyramid was three times greater than that of the smaller pyramid, and none of the other dimensions were different, the volume would be scaled proportionally. Again, it turned out I was correct.

I actually started the third exercise but got a few questions wrong so I didn’t bother finishing it off. I’m not going to go into details of this question, but here’s an example from the exercise:

Exercise 3 – Volume of Prisms and Pyramids

Question 8

I’ll finish off this exercise at the start of this upcoming week and go through some example questions then.

So, that was it for this week. Pretty disappointing but progress nonetheless. I think I should be able to get through the two remaining exercises and unit test from Solid Geometry this upcoming week and will then be moving on to Algebra 2. I’m 96% of the way through that course which means it shouldn’t take too long to get through. There are four units that have unit tests I need to redo, and a handful of exercises between the four of them. I think it’s possible I could get through that course by September, but I’ll have to make faster progress than I’ve been making lately if I want to make it happen. As usual, fingers crossed! 🤞🏼