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1/7 is around 14% I think

I remember this from this years AMC 10 lol

ABC is a right triangle with an angle of C=90 degrees. AC = 10 and BC = 24. Point P is located inside ABC such that the distance from P to AB is twice the distance from P to AC, and the distance from P to AC is equal to the distance from P to BC.  In units, what is the distance from P to AB?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[ABC] = 120

x² + 1/2[x(10-x) + x(24-x)] + 26x = 120

x = 120 / 43

The distance from P to AB = 2x

1/a + 1/b = 1/3

2/a + 1/b = 1/2

$\binom{4}{1, 2, 3, 4}$

cosA = 1 / √2

Melody, thank you for going to the trouble to explain that to me.  I get it.  Your explanation made it easy to understand. 

\(\frac{1}{x}+\frac{1}{y}=\frac{1}{18} \\x+y=\frac{xy}{18} \\18x+18y=xy \\xy-18x-18y=0 \\xy-18x-18y+324=324 \\(x-18)(y-18)=324\)

factors of 324: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, and 324

can you do it yourself now? 

JP

Your linear sequence could be:

2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 , 24 , 26 , 28 , 30 , 32 , 34 , 36 .....etc.

Your exponential (geometric sequence) could be:

2 , 6 , 18 , 54 , 162 , 486 , 1458 , 4374 , 13122 , 39366 , 118098 , 354294 , 1062882 , 3188646 , 9565938 , 28697814 , 86093442 .......etc.

Note: This exponential sequence can be written like this:

2, (2*3^1), (2*3^2), (2*3^3), (2*3^4)........and so on.

You can find all the info you need from 2 sequences. Good luck.

#2 is 53/200

im thinking of another way...what if we say that the sqrt thingy is a variable...the things behind it are the same

P. S. an aops algebra 1(or another book)  has a similar problem

also im bored theres no questions being post :/

WHOA LOOK AT THIS DETAILED SOLUTION

Angle of the vertex of the combined shapes so far (without n sided) = $360 - (90 + 120 + 60) = 90.$ so we need a shape that has angle side 90 to finish the 360 degrees. therefore, it is a square: with 4 sides. it is an n-sided polygon, so n = 4

No prob....I just hope it's correct!!!!   (HAHAHA!!!)

THANK YOU SO MUCH!!!!!!!!!

I had no idea it was that simple...  ELEGANT solution 

Note  that  the  given expression  = 9

Square  both sides

x  +  given expression  =      81

x  +  9    =   81

x   = 81  - 9    =    72

 BIG BRAIN

Hey....that's OK.......at least  we got  the  same answer.......great minds think alike....LOL!!!!!!

OOPS i didnt see this solution when i posted mine! great they are the same though :)

We have P = k(14,7) where P = pressure and k = the rate

at 0 feet is 14.7 lb

calculate, the pressure increases at a rate of 0.441 psi for each foot you descend. 

mile is 5280 feet

so 14.7 + 5280(0.441) = ~2343.2 psi

We  can create  the points  (depth , pressure)   = ( 0 , 14.7)  and  ( 100 , 58.8)

Take  the  slope  between  these points

[ 58.8 - 14.7  ] /  [ 100  - 0  ]   =    44.1  /  100  =     .441

Using the point  (  0 ,14.7)....the  equation for pressure, p,  is

p =   .441d   +  14.7

1 mile  = 5280 ft = d

p   = .441 (5280)   +  14.7  =    2343.18  lbs  /  in^2

The minimum possible value is 70 - 50 = 20%, and MAXIMUM is 30 + 50 = 80%

difference = |80 - 20| = 60

Very nice, JP    !!!!!!