Mathematics Thread

[Stefan]

Yesterday I had a breakthrough moment and, for those who are interested, here is my function for all integer n:

Spoiler

\( f(n)=(n\mbox{ }mod\mbox{ }7)-7\lceil\frac{(n\mbox{ }mod\mbox{ }7)}{7}-\frac{1}{2}\rceil \)

Where \( \lceil n \rceil \) is the ceiling function of n, also called the least integer function.

That's rather complicated. I usually do:

Spoiler

((n + 3) mod 7) - 3

 

[cmhardw]

That's rather complicated. I usually do:

Spoiler

((n + 3) mod 7) - 3

Wow Stefan, that's really clever! For some reason I had never thought to perform an operation on the variable, then examine that result mod a number. Cool!

 

[cmhardw]

Earlier today I saw the question "How many times a day does a clock’s hands overlap?". I thought this included the second hand. So I went and calculated a result. Apparently, just about everyone else on the internet took this question to mean "How many times a day does a clock’s minute and hour hands overlap?". I was correct in that the minute and hour hand overlap 22 times in one day. However, I could not find anyone who calculated the result when the second hand was included. So I am looking to see if some of you math geniuses could confirm or debunk my result on the how many times a day ANY of the hands on a clock (Second, minute, and hour) overlap.

I got your same number.

My method was:

Spoiler

I used a constructive proof by just finding all the exact times that any specific overlap would happen. Rather I found a formula for each pair of hands to find what time the overlaps happen. I've heard of this clock problem before, but I have never tried it myself. This was fun! Basically I approached it from the viewpoint of an angular velocity problem.

The minute hand overlaps the hour hand 11 times in a 12 hour period. There is no overlap of the minute and the hour hand during hour 11 (this overlap happens right on 12:00 exactly).

The second hand overlaps the minute hand 59 times every hour. There is no overlap of both hands between 11:58:58.98 and 12:00 until exactly 12:00.

The second hand overlaps the hour hand 719 times in a 12 hour period. There is no overlap of the second and hour hands from 11:58:59.92 and 12:00 exactly.

24*59+719*2+11*2=2876

However, this has counted 0:00 and 12:00 three times each. On the clock that means I have counted 12:00 six times, when it should only have counted twice.

2876-4=2872, same as you.

Fun problem!

 

[Stefan]

Wow Stefan, that's really clever! For some reason I had never thought to perform an operation on the variable, then examine that result mod a number. Cool!

Fun fact: I couldn't remember the last time I used that, but then I used it again *today*. Came across Gauss's weekday algorithm where I was given the month as 1..12 and had to subtract 2 and end up in 1..12, which I did as ... [exercise for the reader , solution below]

Spoiler: solution

((m - 3) mod 12) + 1

 

[TheNextFeliks]

Maths joke (also in my sig): sqrt(-1) 2^3 Σ π

 

[ben1996123]

Maths joke (also in my sig): sqrt(-1) 2^3 Σ π

\( \frac{i+j+k}{\sqrt{3}} \) ate all the pi?

 

[TheNextFeliks]

\( \frac{i+j+k}{\sqrt{3}} \) ate all the pi?

Huh?

 

[ben1996123]

Huh?

\( \left( \frac{i+j+k}{\sqrt{3}}\right)^2=-1 \)

Q8 group

 

[Smiles]

The curve of a suspension bridge cable attached between the tops of two towers can be modelled by the function
h(d) = 0.0025(d - 100)^2 - 10,

where h is the vertical distance from the top of a tower to the cable and d is the horizontal distance from the left end of the bridge, both in metres. What is the horizontal distance between the two towers? Express your answer to the nearest tenth of a metre.

a) 213.1m
b) 126.4m
c) 138.6m
d) 169.3m



I need to solve this question and I'm pretty sure I would know how, if only the wording made sense. Can anyone help and explain?

 

[CheesecakeCuber]

where h is the vertical distance from the top of a tower to the cable and d is the horizontal distance from the left end of the bridge, both in metres. What is the horizontal distance between the two towers? Express your answer to the nearest tenth of a metre.

Not sure I understand this.

 

[Smiles]

Not sure I understand this.

neither do i. that's the problem :/

EDIT:
nevermind i found this.

Spoiler

 

[ben1996123]

Spoiler

 

[Smiles]

thanks for your help! i managed to do it on my own too.
i have another one D:

Dinahi’s rectangular dog kennel measures 2 ft by 5 ft. She plans to double the area of the kennel by extending each side by an equal amount.
Determine the equation to model the new area.

so what i'm thinking is each side (2 and 5) should have x added to it, and when you multiply them they would equal double the current area (10*2)

so (x+2)(x+5)=2(2)(5)
x^2 + 7x + 10 = 20
x^2 + 7x - 10 = 0 is my answer.

i dont have the multiple choice, but a,b,c all started with 4x^2, and d started with x^2 but was not equal to what i got.
help would be appreciated.

---

there was another question that was similar; the same thing but 4 ft + 7 ft and it asked how much each side should be extended by. i did get the answer but the multiple choice was all different.

edit: read my next post the wording was stupid.

 

[CarlBrannen]

The problem of a chain hanging between two towers is an interesting mechanics problem. The solution isn't quite a quadratic equation, it's a catenary:
https://en.wikipedia.org/wiki/Catenary

According to the wiki article, Galileo thought it was a parabola.

To see that it's not a parabola, you can go through a lot of math to calculate the actual curve, but I suspect that there's a simple argument based on symmetry. I vaguely recall writing up a solution to the catenary problem for a class some years ago...

 

[Smiles]

Dinahi’s rectangular dog kennel measures 6 ft by 8 ft. She plans to double the area of the kennel by extending each side by an equal amount. How much does she extend to each side, to the nearest tenth of a foot?

a. 1.4
b. 3.2
c. 5.7
d. -8.4

---

i didn't get any of those so idk whats up.

(x+6)(x+8) = 2(6)(8)
x^2 + 14x + 48 = 96
x^2 + 14x + 49 = 97
(x+7)^2 = 97
x+7 = √(97)
x = -7 + √(97) // not minus or it'll be negative
x = 2.8488578...

and i even tested it. adding it onto each side and multiplying it should give me double of 6*8, which is 96.
(6+2.8488578)*(8+2.8488578) // approximately
= (8.8488578)*(10.8488578)
= 96

edit! OMG I JUST REALIZED it means extend each of the 4 directions, (front back left right) not just each actual side (wall) of the kennel. i hate the wording of this stuff i can't believe i didn't notice that.

i no longer need help on these questions, clearly my brain was dead for the day.

 

[ben1996123]

Dinahi’s rectangular dog kennel measures 6 ft by 8 ft. She plans to double the area of the kennel by extending each side by an equal amount. How much does she extend to each side, to the nearest tenth of a foot?

okso (2x+6)(2x+8)=96
x^2+7x-12=0
x=(-7+sqrt(97))/2 wich is about 1.4244289008980523608731057074588



okso whast \( \frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x^2 \)

 

[CarlBrannen]

I also interpreted the extension as "two sides" because that's what any carpenter would do.

An easy way of doing the problem:

The area is 48 square feet. The perimeter is 6+8+6+8 = 28 feet. So the answer is going to be a little less than 48/28 = 24/14 = 12/7 = 1 5/7 = 1.52 feet. But only one of the answers is less than 1.52 feet.

 

[brandbest1]

I'm assuming LaTeX works perfectly fine on speedsolving.

\frac{\pi}{2}

Someone help me, please?

 

[Stefan]

I'm assuming LaTeX works perfectly fine on speedsolving.

\frac{\pi}{2}

Someone help me, please?

Look at how others do it (which you can see when you quote them).

 

[brandbest1]

Look at how others do it (which you can see when you quote them).

\( Thanks, ~ Stefan! \)