Presumably easy sub-question to a tricky calc 1 problem

[LearninDaMath]

Okay so the question involves finding the dimensions of a container of least cost given a specified volume and the cost of material on various surfaces.

However it gets down to this one part where I have a number 2250. In my notes, my teacher automatically converts 2250 to (15^2)(10) as a simplification.

My question is how does my teacher make this leap? There are no calculators allowed in this course so is there some method to this arithmetic madness or do you just have to be a human calculator?

 

[Mark44]

Okay so the question involves finding the dimensions of a container of least cost given a specified volume and the cost of material on various surfaces.

However it gets down to this one part where I have a number 2250. In my notes, my teacher automatically converts 2250 to (15^2)(10) as a simplification.

My question is how does my teacher make this leap? There are no calculators allowed in this course so is there some method to this arithmetic madness or do you just have to be a human calculator?

2250 is obviously divisible by 10 because of the 0 digit, which leaves the other factor as 225, which is the square of 15. You do know your times table up to 15 x 15, right?

 

[LearninDaMath]

2250 is obviously divisible by 10 because of the 0 digit, which leaves the other factor as 225, which is the square of 15. You do know your times table up to 15 x 15, right?

Up to 15 x 15? Since when did they start teaching times tables up to 15 x 15?

When I was in elementary, they only went up to 12 x 12 ...and didn't even emphasize actually remembering the 12s. What school/district/city/state actually covers up to 15 as a standard practice?

P.S. Thanks for the response, much appreciated.

 

[eumyang]

Up to 15 x 15? Since when did they start teaching times tables up to 15 x 15?

When I was in elementary, they only went up to 12 x 12 ...and didn't even emphasize actually remembering the 12s. What school/district/city/state actually covers up to 15 as a standard practice?

When I was in elementary school we only went up to 10x10. Back then I had learned up to 12x12 on my own.

But anyway, even if you didn't learn the times tables up to 15x15, you should have at least learned the 1st 20 perfect squares (1² to 20²), if not more.

 

[LearninDaMath]

When I was in elementary school we only went up to 10x10. Back then I had learned up to 12x12 on my own.

But anyway, even if you didn't learn the times tables up to 15x15, you should have at least learned the 1st 20 perfect squares (1² to 20²), if not more.

Squares up to 20? Wow, you guys are making me feel somewhat cheated by the educational system of my younger days. I remember being in an F school in the 48th lowest performing state, but I don't remember anyone learning exponents until around 6th grade. And we were never tasked to actually memorize square powers - the concept of exponents were explained and we just used what we already knew from our (limited, i guess) times tables (2x2,3x3, etc).

Glad I woke up a couple years ago and realized how significant and rewarding math (and science) can be. I've been working hard in the time available to learn as much as I can. It's really tough learning all this stuff while not having the fundamentals already deeply engrained, but the synapses are a firing and I'm getting the concepts little by little.

Went from college algebra to Calc over the past year and I'm continously reminded about how few of the basics I really actually know.

Just thankful for the internet and sites such as this. If ever there was a good time in history to learn some math and science, it would be these days.

 

[Mark44]

Really I was being facetious about memorizing the times table up to 15 by 15. I know we did the multiplication table up to 12 times 12, and over the years I have memorized the squares up to 20 x 20 or so.

If you are motivated to put some effort in, that will take you a long way. If you run across problems that use some knowledge that you're rusty on, take a little extra time and review that material. After you've been through it once, going through it a second time is a lot easier.

 

[Deveno]

even if you didn't know that 225 = 15*15, it's still easy to factor 2250:

2250 = 225*10 (obvious).

225 ends in 25, must be divisible by 25...4 25s to the hundred, we have 9 (4 + 4 and 1 left over).

so 225 = 9*25 = 3*3*5*5 = 3*5*3*5 = 15*15 (most of this you can do in your head).

 

[HallsofIvy]

Of course, immediately recognizing that $225= 15^2$ is a matter of experience. However, anyone who has taken arithmetic should be able to recognize that 225 ends in a 5 and so is divisible by 5: 225= 5(45). 45 also ends in 5 and so is divisible by 5: 45= 9(5) so $225= 9(5)(5)= 3^2(5^2)= 15^2$.

 

[Deveno]

Okay so the question involves finding the dimensions of a container of least cost given a specified volume and the cost of material on various surfaces.

However it gets down to this one part where I have a number 2250. In my notes, my teacher automatically converts 2250 to (15^2)(10) as a simplification.

My question is how does my teacher make this leap? There are no calculators allowed in this course so is there some method to this arithmetic madness or do you just have to be a human calculator?

it is also quite likely that at some point your teacher has had some exposure to number theory, which is devoted to the recognition of certain patterns that occur in numbers. with even a small amount of time doing such things, certain patterns (squares, cubes, differences of squares, and prime factorizations) become a sort of second nature.

i daresay even you would be able to display such "amazing" powers, given the exposure to the right subjects.

 

[LearninDaMath]

Really I was being facetious about memorizing the times table up to 15 by 15. I know we did the multiplication table up to 12 times 12, and over the years I have memorized the squares up to 20 x 20 or so.

If you are motivated to put some effort in, that will take you a long way. If you run across problems that use some knowledge that you're rusty on, take a little extra time and review that material. After you've been through it once, going through it a second time is a lot easier.

Well Mark, I'm feeling very motivated to spend some effort learning some of those higher order times tables and exponents. I would venture to bet that an astute individual who had the foresight to remain attentive throughout his school days would come to recognize these larger basic calculations by heart as a matter of direct practice or indirect experience as HallsOfIvy said. I got to say i am envious of such "amazing" powers at this point in time lol.

it is also quite likely that at some point your teacher has had some exposure to number theory, which is devoted to the recognition of certain patterns that occur in numbers. with even a small amount of time doing such things, certain patterns (squares, cubes, differences of squares, and prime factorizations) become a sort of second nature.

i daresay even you would be able to display such "amazing" powers, given the exposure to the right subjects.

yea, I am more than slightly convinced my teacher is a mathematical genius of some sort. My school's webpage that contains my teacher's syllabus also contains a link to his curriculum vitae and his 30 plus professional publications, funded research, invited lectures..and the list goes on and on and on. and...i might say...it's intimidatingly impressive. So I wouldn't be suprised if he's has dabbled in number theory at some time.

 

[LearninDaMath]

Quick update and analysis... it seems like half my current calc1 class has dropped. Of us who are still there, some are doing really well and most others are doing really poorly. It seems that those who are doing really well have taken calc for at least a year in high school or are repeating the current college calc class. Those who are doing from "okay to"relatively well" may not have a strong calc background, but have at least a really strong algebra and trigonometry background. Of those doing poorly, some have a strong algebra and trig background, but the calculus just hasn't "clicked" yet...and then there are those people who are getting most of the calculus concepts, but whose algebra and trig background is lacking enough to hinder the ability to get through complex calc problems on tests & quizes, most of which involve a boatload of algebra and trig.

I fall into the last category. I'm not doing very well at this point in the class. I get the concepts, but each homework/example problem, quiz problem, test problem, contains some amount of "tricky" algebra and trig for which I have to go back and take some time to learn prior to actually getting through the relevant calc problem at hand. That there seems to be my downfall.

But I go to tutoring and many times, the hw/review/class notes are so confusing that the tutors at the university have a tough time deciphering them and sometimes I just have to accept the fact that the only person who knows what the professor is talking about is the professor himself..lol.

Anyway, attached to this post is the original problem for which I started this thread. I rewrote my class notes in a little more detail so it could hopefully be easier for anyone who may have some trouble with this or a similar problem in the future. BTW, Please note that had I thought about posting these notes when i wrote them, I'd have probably tried to write somewhat more neatly, and also note that when I am rewriting class notes for myself, I usually write "we" to refer to me in the present instructing my future self on how to do the problem lol

 

[LearninDaMath]

P.S. I just noticed the sticky at the top of this section regarding not posting textbook style questions. I'm guessing this post could be considered textbook style. Just that when I started the thread, it didnt seem to be a textbook type of question at that point in time.