Let’s say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x²+y²=1$).
It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.
My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I’m not looking for a series of algebraic equations but rather for some kind of a visual proof).
Question: Activities to do in high school that you can join without trying out?
My freshman year of high school just started, and I’m pretty bored. I can’t do sports because I was traveling during try-outs. I tried out for the play and the dance team, but I didn’t make either. Do you have any ideas for something else I could do outside of school?
Question: Why am I going to be a failure in life for not doing well in school and not going to college?
You got that right, I’m not doing well in school and I’m deciding to not go to college. Because why should I? Just because society said so? Why can’t society understand that the education in school is outdated and grabage? Why can’t society understand that there are lots of opportunities in the real world?
And I’m not saying what I’m doing after I’m finished with school, but I can say that I’m going to be better off than those who went to college and got good grades in school?
Why is society so stupid to those who are different??????
Question: For high school what is the point of studies or resource rooms?
I know studies and resource rooms are meant to finish homework but I always finish my homework before a study so I have to sit for a whole hour doing absolutely nothing no phones no sleeping just sitting doing nothing why