222 replies to this topic

[Secret Igshar]

ill scan my work to show these buggers how its done

I held your hand through the whole thing. Don't act all high and mighty. :>

HRM. Lemme get something below Calc to see if others can get it.

Jimmy the Alligator is trying to build a fence to lock up his human meatsnacks for later. He has 100 meters of fencing to make this fence and is building it alongside his river, so he has easy access to his snacks, so he doesn't want a fence along the riverside. He wants to hold as many humans in his enclosure as possible. What should be the dimensions of his enclosure so as to maximize the storage space for human meatsnacks?

Assume every human meatsnack has already been bound and gagged by Jimmy's slave monkey Albert.

[Secret Felix]

___50___
25| |25
~~~~~~~~~~

Maybeh.

[Secret Igshar]

___50___
25| |25
~~~~~~~~~~

Maybeh.

I forwarded the answer to Jimmy and he told me that you get to be one of the lucky humans who get to cross the river without being bound and gagged by Albert.

Somebody else needs to make up a more difficult problem. D:

[Lexaeus]

Let me fuel the fire...

1.) An insect lives on the surface of a regular tetrahedron with edge of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite of they have no common endpoint.)

2.) Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What is the units digit of N?

3.) Let f be a function satisfying f(xy) =f(x)/y for all positive real numbers x and y. If f(500) = 3, find f(600).

4.) A line passes through A(1,1) and B(100,1000). How many other points with integer coordinates are on the line and strictly between A and B?

Show your work. >:|

[Secret Igshar]

1.) An insect lives on the surface of a regular tetrahedron with edge of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite of they have no common endpoint.)

HURM. I believe this distance would be to travel from the midpoint along a face to the opposite vertex (1/sqrt(2) units) and then halfway along the opposite edge (0.5 units)

Since I don't have a calculator on me at current, I'mma just say my answer is

(2 + sqrt(2)) / 2

And got a calculator to get a decimal of about 1.707 units.

2.) Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What is the units digit of N?

9, dur. Dunno how to show work for it, but every N with X9 will satisfy that.

3.) Let f be a function satisfying f(xy) =f(x)/y for all positive real numbers x and y. If f(500) = 3, find f(600).

Let xy=500.

Thus:

y = 500/x

And:

3 = ( x f(x) ) / 500

If x = 600 , then y = 5/6

Therefore:

3 = ( 600 * f(600) ) / 500

3 = (6/5) * f(600)

2.5 = f(600)

<3

4.) A line passes through A(1,1) and B(100,1000). How many other points with integer coordinates are on the line and strictly between A and B?

Show your work. >:|

Well, from Slope, you can gather that the slope is 111/11, so you'll have 8 points in-between and on the line. I can list them if you'd like.

[Lexaeus]

HURM. I believe this distance would be to travel from the midpoint along a face to the opposite vertex (1/sqrt(2) units) and then halfway along the opposite edge (0.5 units)

Since I don't have a calculator on me at current, I'mma just say my answer is

(2 + sqrt(2)) / 2

And got a calculator to get a decimal of about 1.707 units.

Try this one again pl0x.

5.) Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set (A1, A2, …, A9)?

6.) How many perfect squares are divisors of the product 1!*2!*3!...9!?

7.) Let f(x) = sq(ax^2+bx). For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the same set?�3!􀂘􀂘􀂘9!?*33

[Secret Igshar]

Try this one again pl0x.

It's 1 unit. :> From midpoint to midpoint to midpoint.

5.) Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set (A1, A2, …, A9)?

By in the plane of, do you mean within the polygon or actually on its plane? If it's on the plane, there's lots and lots. 18 just by using two that are next to each other. If it's just within the polygon, the number is much smaller. Clarification plox.

6.) How many perfect squares are divisors of the product 1!*2!*3!...9!?

Well it'd be also written as:

9¹*8²*7³*6⁴*5⁵*4⁶*3⁷*2⁸*1⁹

right? That's a lot of 2's... To simplify it...

3²*2⁶*7³*2⁴*3⁴*5⁵*2¹²*3⁷*2⁸

And~ Combining like terms yields:

7³ (7²)
5⁵ (5² and 5⁴)
3¹³ (3² to 3¹² [Even powers])
2³⁰ (2² to 2³⁰ [Even powers])

so:
1
2
6
15

So this starts us off with a base amount of perfect squares of
1+2+6+15=24

Adding in the number 1~ gets us 25 as our base.

Now for any combination of them. If there are only two values added into the perfect square:

1*2=2
1*6=6
1*15=15
2*6=12
2*15=30
6*15=90

2+6+15+12+30+90 = 155

If there are 3 numbers in each:
1,2,6,15

1*2*6=12
1*2*15=30
1*6*15=90
2*6*15=180

12+30+90+180 = 312

And if there are 4 numbers in the perfect square:

1*2*6*15=180

So we get:

25+159+312+180 = 672 distinct, possible combinations of perfect squares.

7.) Let f(x) = sq(ax^2+bx). For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the same set?�3!������9!?*33

Exactly 1. If a=0, b can be any positive integer and cause the range and domain of f to be ( 0 , ∞ ). As long as the x² term exists in there, there will be both positive and negative values within the domain of the function, but the range will never be allowed to go below zero.



What is the sum of every number from 1 to 1,000,000?

[Lexaeus]

By in the plane of, do you mean within the polygon or actually on its plane? If it's on the plane, there's lots and lots. 18 just by using two that are next to each other. If it's just within the polygon, the number is much smaller. Clarification plox.

Within. Try again or let someone else have a go.

What is the sum of every number from 1 to 1,000,000?

S=n(l+f)/2
=1000000(1000000+1)/2
=500000500000

8.) A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050. If the odometer now reads 002005, how many miles has the car actually traveled?

9.) For how many ordered pairs of positive integers (x,y) is x + 2y = 100?

10.) A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of the box. What percent of the original volume is removed?

Edited by Lexaeus, 21 February 2010 - 12:54 AM.

[Secret Igshar]

Within. Try again or let someone else have a go.

Too many. Meh.

8.) A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050. If the odometer now reads 002005, how many miles has the car actually traveled?

Lessee. Every ten miles, it skips 1. Every 100 miles, it'd skip 1 (in addition to the 9 skipped from each set of 10)

This means every 10 = 9 and every 100 = 81, down 10 due to skipping 40, 9 from skipping x4 9 times.

Every 1000 would therefore be down 19x9+100 = 271 so 2000 would be down 542. Since it's 2005, it skipped another 4, so it's down 543, putting it at 1462 miles driven.

9.) For how many ordered pairs of positive integers (x,y) is x + 2y = 100?

Lots and lots and lots. Every even integer of x from 2 up to 98 would give out one solution.

I'm gonna say that you'd have 49 different ordered pairs of positive integers.

10.) A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of the box. What percent of the original volume is removed?

( 3³*8 ) / ( 15x10x8 ) = 18%


What area is beneath the graph of y=cosx from 0 to 2pi?

[Secret Felix]

Water is leaking out of a conical tank at a rate of 12,000 cm³/min at the same time as water is being pumped in at a constant rate. The tank has a height of 8 m and the diameter of the top is 6 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 4 m, how fast is water being pumped into the tank?

Not sure on this answer, since I only have the notes left of where I worked it~

[Secret Igshar]

Uh. I got 5,666,866.776 ^cm3/_min.

Work was looooong, but I started with:

V = ¹/₃πr²h

^dV/_dt = x_in - x_out

^r/_h = ³/₄

h = 400 cm
x_out = 12,000 ^cm3/_min
^dh/_dt = 20 ^cm/_min

[Deadly-Dreamer-X]

How to find the circumference of an Ellipse?

I only know the dimensional formulas(Chapter named conic section, something like x1/a^2 + x2/b^2), but not the metrical measurement and what no.

[Secret Igshar]

How to find the circumference of an Ellipse?

I only know the dimensional formulas(Chapter named conic section, something like x1/a^2 + x2/b^2), but not the metrical measurement and what no.

It's an extremely complicated process, involving integrals. >_>; Eh. You also have that wrong. =X

Ellipses are circles where x² and y² have different coefficients. In general, the equation of a circle is:

x² + y² = r²

This can also be written as:

^x2/_r² + ^y2/_r² = 1

For an ellipse, the general equation is given by:

^x2/_a² + ^y2/_b² = 1

where a is the "horizontal radius" and b is the "vertical radius"


The circumference is...

4aE(e²)

where e is the eccentricity and is equal to:

e = sqrt( 1 - (^b/_a)² )

and E is something called an Elliptical Integral, which >_> I don't really understand, so I can't help with that one. Basically, it's an integral that doesn't work out too nicely and needs to be solved with infinite series.


All of this adds up to say that you should probably not bother finding the circumference of an ellipse. <_____<;;



Oh joy. I had forgotten that conics are gotten by intersecting a cone with a plane. I have a bad feeling we're gonna be going into them in Calc 3. >_>;

OH OH. Here are a couple of questions from my Calc 3 test yesterday! I wonder if anybody can solve them~


The temperature at any point (x,y) on a steel plate is given by T = (y) / (x² + y²) where x and y are measured in meters. At the point (2,1), find the rate of change of the temperature with respect to the distance moved along the plate in the y direction.


Suppose w = sqrt( x² + y² ), with x = sin2t and y = 3e^tcos2t. Find the value of ^dw/_dt when t = 0.

I was supposed to use the Calc 3 Chain Rule for that^ :O

I picked the two... "easiest" ones, since they just involve differentiation.

[Deadly-Dreamer-X]

I still have trouble with the principle of Mathematical Induction.

I am always stuck right after P(K+1) =
Don't know what to do next.

[Secret Igshar]

I still have trouble with the principle of Mathematical Induction.

I am always stuck right after P(K+1) =
Don't know what to do next.

Well Mathematical Induction is just where you prove that P(k) is true and then prove that P(k+1) is true for the arbitrary k. You then just need to prove that P(0) is true which would then prove that P(1) is true and that P(2) is true, et cetera, proving up eventually that P(n) is true for all whole numbers.

*shrugs* What don't you get? o-o;

[Deadly-Dreamer-X]

What is true? What is proving? What hell!

see, let me do something crazy here

Here is a question.


1+2+3.....+n = n(n+1)/2

Let say I finished it till this step,

P(k+1) = k+1 = (k+1) (k+2)/2

I am stuck here..I don't know the next step, and the following step and to where I get to see L.H.S of what I am doing.

[Secret Felix]

Once upon a time there was a clockmaker named Wynn. One day he decided to train his clock-making skill. Wynn begins at level 1 with 1,347,821 experience. In order to advance to a higher level of clock-making, he decides to make rusty watches. Each rusty watch gives 651 xp, and requires 166 cogs. Upon reaching level 3, each rusty watch can be disassembled for 117 cogs. Upon reaching level 10, Wynn gains the ability to make 5-second-slow watches. Each of these watches give 747 xp while requiring 150 cogs. Upon reaching level 12, Wynn learns he can disassemble each 5-second-slow watch for 105 cogs. Assuming that Wynn is always making the highest level watch possible, please find:

The number of rusty watches to reach level 10.
The number of 5-second slow watches to reach level 15.
Assuming that all watches are taken apart and reused, the lowest number of cogs needed to reach level 15.

[spoiler=Level/Experience Table]
[table]1 1,336,443
2 1,475,581
3 1,629,200
4 1,798,808
5 1,986,068
6 2,192,818
7 2,421,087
8 2,673,114
9 2,951,373
10 3,258,594
11 3,597,792
12 3,972,294
13 4,385,776
14 4,842,295
15 5,346,332[/table][/spoiler]

Good Lucky~

[Secret Igshar]

To get to level 10, Wynn would require 2,936 Rusty Watches.

To get to level 15 from there, he would need 2,795 Two-Second-Slow Watches.

Altogether, he would need 340,514 Cogs, assuming he only has time to melt the Watches back into Cogs upon each level up or at the end of the level. :>

[Lexaeus]

Let's try some algebra...

a=b

multiple by a

a^2=ab

add a^2

a^2+a^2=a^2+ab

simplify

2a^2=a^2+ab

subtract 2ab

2a^2-2ab=a^2+ab-2ab

simplify

2a^2-2ab=a^2-ab

factor out (a^2-ab)

2(a^2-ab)=1(a^2-ab)

divide by (a^2-ab)

2=1

Math gone craZy? Or did I error? If so, where and why?

[Secret Igshar]

a^2=ab

divide by (a^2-ab)

Derp. Same reason you cannot divide by x or y when doing equations when the variable in question can be zero.

Anyway, find me the line parallel to:

x = 2 + 3t
y = 4 + 6t
z = 1 + 2t

that passes through the point (3,5,7)

Obviously it's in 3D, not 2. :> ENJOY.