2012 Year 9 &10 Mathematics Marathon (5 Viewers)

[SpiralFlex]

Re: 2012 Year 9 &10 Mathematics Marathon

How many jugs are we allowed to have?

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Fill 5L with water and use it to fill the 3L. You have 2L left.

Pour out 3L and put the 2L into the 3L jug.

Fill the 5L again and fill the remainder (1L) of the 3L jug, leaving 4L.

 

[enoilgam]

Re: 2012 Year 9 &10 Mathematics Marathon

Fill 5L with water and use it to fill the 3L. You have 2L left.

Pour out 3L and put the 2L into the 3L jug.

Fill the 5L again and fill the remainder (1L) of the 3L jug, leaving 4L.

Thats it. The problem is from the movie Die Hard: With a vengence - carrots method was the way it was solved by Lt McClane and Zeus.

 

[SpiralFlex]

Re: 2012 Year 9 &10 Mathematics Marathon

So are we allowed to post up logic questions?

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Thats it. The problem is from the movie Die Hard: With a vengence - carrots method was the way it was solved by Lt McClane and Zeus.

Haha I never knew, interesting that a Hollywood film would use a Maths problem....

 

[enoilgam]

Re: 2012 Year 9 &10 Mathematics Marathon

Haha I never knew, interesting that a Hollywood film would use a Maths problem....

You better have seen die hard. In the movie, they need to solve it in order to disarm a bomb.

 

[Fawun]

Re: 2012 Year 9 &10 Mathematics Marathon

Fill 5L with water and use it to fill the 3L. You have 2L left.

Pour out 3L and put the 2L into the 3L jug.

Fill the 5L again and fill the remainder (1L) of the 3L jug, leaving 4L.

Hey last time i checked this was a "Year 9 and 10" maths thread

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Hey last time i checked this was a "Year 9 and 10" maths thread

Well nobody was getting it the 'intended way' so I thought I may as well give it a shot.

Another question:

Carrot can jog at 15km/hr for X minutes whereas Fawun can sprint at 5km/hour for an indefinite amount of time.

They both start at the same time on a 400m oval track.

How long should Carrot be able to jog at this speed, such that he can overlap Fawun whilst maintaining that speed?

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Oh and here's a cool Geometry one.

Pick an arbitrary point on an equilateral triangle.

From this point, perpendiculars from each side are constructed to meet at that point.

Prove that the total length of these perpendiculars is the same as the length of the altitude.

 

[Fawun]

Re: 2012 Year 9 &10 Mathematics Marathon

Well nobody was getting it the 'intended way' so I thought I may as well give it a shot.

Another question:

Carrot can jog at 15km/hr for X minutes whereas Fawun can sprint at 5km/hour for an indefinite amount of time.

They both start at the same time on a 400m oval track.

How long should Carrot be able to jog at this speed, such that he can overlap Fawun whilst maintaining that speed?

is this a srs question

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Actually, yes.

 

[chocolatelatte]

Re: 2012 Year 9 &10 Mathematics Marathon

Oh and here's a cool Geometry one.

Pick an arbitrary point on an equilateral triangle.

From this point, perpendiculars from each side are constructed to meet at that point.

Prove that the total length of these perpendiculars is the same as the length of the altitude.

Let the altitude of the equilateral triangle be h, and the sides be x.
then the area of this triangle is xh/2.

when you construct the perpendicular lines from an arbitrary point, you get 3 triangles all with equal bases (x).
let the height of each triangle be a, b, c (order isn't rly important here).
then you add up the areas of the 3 triangles so you get an alternate value for the area of the equilateral triangle.
since they're equal you get an equation

ax/2 + bx/2 +cx/2 = hx/2

>>> simplify and you get x(a+b+c) = x(h)
cancel the x and you get a+b+c= h

Well nobody was getting it the 'intended way' so I thought I may as well give it a shot.

Another question:

Carrot can jog at 15km/hr for X minutes whereas Fawun can sprint at 5km/hour for an indefinite amount of time.

They both start at the same time on a 400m oval track.

How long should Carrot be able to jog at this speed, such that he can overlap Fawun whilst maintaining that speed?

i did it but i'm not sure if my answer's correct cos I didn't really take into account the 'x' + 'indefinite amount of time' (is that even relevant? i don't understand).

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Good solution for Viviani's Theorem: http://en.wikipedia.org/wiki/Viviani's_theorem

=)

As for the running one, it's not really needed, just wanted to specify that the speed was maintained indefinitely to avoid people thinking she would tire etc.

You don't really need to use X, just need to calculate the time it takes to overlap.

 

[iBibah]

Re: 2012 Year 9 &10 Mathematics Marathon

Well nobody was getting it the 'intended way' so I thought I may as well give it a shot.

Another question:

Carrot can jog at 15km/hr for X minutes whereas Fawun can sprint at 5km/hour for an indefinite amount of time.

They both start at the same time on a 400m oval track.

How long should Carrot be able to jog at this speed, such that he can overlap Fawun whilst maintaining that speed?

Nice question

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Consider a circle of radius 1.

A circle concentric to that one is constructed such that it is larger.

What should the radius of this larger circle be, such that the 'area difference' is the same as the area of the original circle?

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Suppose a stick were placed in the 'gap' between those two circles.

What is the longest length that the stick can be?

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

The stick divides the grey region into 2 separate areas. What is the area of the smaller one?

 

[RealiseNothing]

Re: 2012 Year 9 &10 Mathematics Marathon

Consider a circle of radius 1.

A circle concentric to that one is constructed such that it is larger.

What should the radius of this larger circle be, such that the 'area difference' is the same as the area of the original circle?

Just a cool way of doing it:

Construct a circle with radius 1.

Construct a square such that the circle touches the midpoint of each of the sides of the square.

Construct a second circle such that the square is concyclic.

Now you have your circle of radius one, and the larger concentric circle.

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

Consider a circle of radius 1.

A circle concentric to that one is constructed such that it is larger.

What should the radius of this larger circle be, such that the 'area difference' is the same as the area of the original circle?

Apologies for not seeing this question earlier:

I let the radius of the larger circle be x and the radius of the smaller circle be r,



Rearrange for x and eventually you get:



Therefore, we are given that the radius of the circle is a unit, therefore the radius of the larger circle is units.

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

Suppose a stick were placed in the 'gap' between those two circles.

What is the longest length that the stick can be?

Is the longest length of the stick = 2 units?