Maths quiz

[AVN]

Hitro,

You're absolutely correct.
It's your turn.

 

[Hitro]

Okay, so why not another series of numbers:

Which number is following after:

6, 24, 54, 96, 150, 216, ?

please give a short explanation for this as well.

 

[gonzo_for_civ]

the formula is n^2 * 6

for example 1 squared * 6 = 6 and 2 squared * 6 = 24

so 49 * 6 = 294

Is this right?

 

[Hitro]

Correct gonzo!

So it's your turn!

 

[gonzo_for_civ]

Ok, here's a little break from the number sequences. Here is an easy trigonometry problem. And I mean easy!

Find the measure of angle a, the length of c, and the length of the hypotenuse b in exact form. Hint: This is a "Special" triangle.

NOTE: I labeled it wrong, the angle that is 30 degrees really should be the top angle and you need to find the bottom right angle. Sorry!

 

[ainwood]

Well, the angle is 60°.

b = 7/sin(60°) = 8.083 cm

c = 7/tan(60°) = 4.041 cm

 

[gonzo_for_civ]

Originally posted by ainwood
Well, the angle is 60°.

b = 7/sin(60°) = 8.083 cm

c = 7/tan(60°) = 4.041 cm

Nope, the angle was right but the legs weren't. Did you get my note about the mislabeling? It's in my post above.

 

[AVN]

Gonzo,

It's strange, but i get the same results as Ainwood.

And I don't see what i'm doing wrong.

 

[gonzo_for_civ]

OK, here goes.

First off, you can solve this immediately using this method, as I said, this is a "Special" triangle.


In a 30-60-90 triangle. The length of the shortest leg is exactly 1/2 of the hypotenuse and the length of the other leg is the length of the smallest leg times the square root of 3.

Basically:

Short leg = L
Hypotenuse = 2L
Long leg = L * sqrt(3)

Or you can find the tan of 60 degrees witch is .5 which is the lengh of the hypotenuse over the adjacent which supports my method.

SO, the answer is thisin exact form)

a = 60degrees
b = 7/sqrt(3)
c = 2(7/sqrt(3)) <-- note the exact form


So, why were you wrong? I asked for exact form

I was just seeing if anyone would catch it

Ainwood, it's yours.

P.S. Don't worry, I wont do that to you in the future

P.S.S. Bet ya learned something new

 

[ainwood]

Originally posted by gonzo_for_civ
SO, the answer is thisin exact form)
P.S.S. Bet ya learned something new

Yep! read the question properly, and watch out for Pedants!

New question:

Take a square, with each side having a length of 100 cm.
Draw a line from each of the internal corners, that goes from that corner to the middle of the opposite side. Do this on each corner, and you should end up with an octagon in the middle. What is the perimeter of this octagon. Show your working.

 

[Lucky]

My result would be 15.31cm (rounded).
I´ll explain if I´m right!

 

[AVN]

Here is my try.

The length of one of the initial drawn lines is :
SQRT(100^2+50^2)=111.8

The length of one of the sides of the octagon is 1/6 of the length of that line. 1/6*111.8=18.63

The perimeter is then 8*18.63=149.07 (rounded)

 

[Lucky]

I made a small mistake, I used 10cm as my square side length.

So my result should really be: 153,1cm.
Sorry.

 

[ainwood]

My answer is different to both of those.....


(Hopefully I haven't stuffed it up)

 

[Lucky]

After reviewing this problem and spending more than just 2 minutes on it, I have to agree with AVN on the results. I used a different approach and get the same results!

Here goes:

1. Use the yellow rectangular triangle to calculate the angle a and b!
2. Use a to calculate the length of the small cathete in the dark yellow triangle!
3. Use Pythagoras to calculate the small cathete in the purple triangle!
4. Use b to calculate one inner angle in purple!
5. Use those to calculate the hypothenuse in purple and take it times 8!

Those are my steps. Here is the actual calculation.

a=side length of square=100cm
other variables not really interesting

1. a= arctan(a/2a)=arctan(1/2), b=90°-a
2. b=a/2*sin a
3. c=SQRT((a/4)²-b²)
4. g=180°-2b=2a
5. d=c/cos b


-> d=(SQRT((a/4)²-(a/2*sin(arctan(1/2)))²)) / (cos(2*arctan(1/2))))

using the value for a we get:

-> d=SQRT(625-500) / 0.6 = 18.6339cm

=> P=8*d=149.0712cm


So you´d better recheck your own calculation, 2x the same exact result would be a huge coincidence.

 

[ainwood]

I redid mine, found a mistake.

Yep, it is 149.1 cm.

I did mine by working out the small angle that lucky shows as "alpha". I then used this to find the length of the line from the edge of the square to the vertice of the octagon. You can then work out the lenght of one of the sides of one of the isoceles triangles that makes up the octagon. The small angle is 45°, and then its straight-forward to calculate the area.

(I had stuffed-up in one of my degrees to radians calcs ).

Its got to go to AVN, although Lucky's proof is a lot more rigorous. Not sure on AVN's statement that the length of each side is 1/6 the length of the original line - obviously it works though!

 

[AVN]

Ainwood,

Thank you for giving the turn to me.
I have to confess that i wasn't 100% sure when telling that one side of the octagon is equal to 1/6 of the original line.
So i was bragging a little bit and this time i was "lucky"

An easy question this time, so to give some newbies to this thread a chance.

A person is travelling from point A to point B with an average speed of 120 km/h. In the afternoon he drives back from B to A with an average speed of 60 km/h.
What is his average speed over the whole day ?

 

[Zwelgje]

I am considering the person is travelling the whole time so there is no break in there in which he doesn't travel.
From A to B he drives 120 km/h, from B to A he drives 60. One would be tempted to add the two and divide it by two but that won't give the right answer (then it would be 90) as the trip from B to A takes twice as long.
Therefore the correct answer should be (120 + 2*60) / 3 = 80 km/h.

 

[AVN]

Civ1_addict,

You are completely right, you did not only find the solution but you also explained why the "intuitive" answer 90 km/h failed.
I say intuitive, because it's my experience 90% of the people will give that incorrect answer.

Good job
and therefore it's your turn now to ask a question.

 

[Zwelgje]

There is a population of 10,000 people and you take a group of 100 people from this population and interview them. Of these 100 people 10 say that they are smokers.
What is the minimum number of smokers in the population of 10,000 people?