CPhill

B. 4

The cut must be a diagonal cut if two triangles are formed

The 13 inch cut  forms the hypotenuse of aright triangle....and the length of the paper forms one of the legs

The width is given by

√ [ 13^2 - 9^2 ]    =  √ [ 169 - 81 ]  =   √ 88   ≈  9.38 inches

Ah, yes.....disregard my answer....!!!

Given: Parallelogram MNPQ, M(3a-b,b) N(3a+b,3b) P(6a+4b,3a+3b)

Find: The coordinates of point Q

The opposite sides are parallel

So....the slope between  MN is 

[ ( 3b - b) ]                           2b

_________________  =   _____  

[ (3a + b) - ( 3a - b)]            2b

And the slope joining PQ is the same....let (x,y) be the coordinates of Q

[ (3a + 3b)  - y]                      2b

__________________ =   ______ 

[ (6a + 4b) - x ]                      2b

This implies that  x = (6a + 2b)  and y = (3a + b)

So Q =   [ 6a + 2b , 3a + b ]

We can check this..since NP and MQ are parallel.....the slope of NP should = the slope of MQ

NP 

[ (3a + 3b) - 3b]                    3a                    a

_________________ =     _______  =      ____

[(6a + 4b) - (3a + b)]           3(a + b)            a + b

MQ

[ b - ( 3a + b) ]                          - 3a         -3a                      a

_________________  =        _____ =   ________   =    _____

[ (3a - b) - (6a +2b) ]             -3a -3b      -3 ( a + b)          a + b 

Note that BC = AC = CD

So......in triangle ACD, angle ACD = angle ACB + angle BCD=  60° + 90°  = 150°

And since  AC = CD.....then the angles opposite these sides in triangle ACD are also equal

So....angle  CAD = angle CDA

So....

angle CAD + angle CDA + angle ACD = 180

angle CAD + angle CAD + angle ACD =  180

2angle CAD + 150  = 180

2angle CAD = 180 - 150

2angle CAD = 30          divide both sides by 2

angle CAD = 15°

-4 -7m < 8 -5m        add 7m to both sides, subtract 8 from both sides

-12 < 2m            divide both sides by 2

-6 < m

Which is the same as

m > -6

16/n^2 * 2.352 = 425.32

Divide both sides by 2.352

16/n^2 = 425.32 / 2.352    and we can write

n^2 /16 = 2.352 / 425.32    multiply both sides by 16

n^2 =   16*2.352 / 425.32     take both roots 

n = ±√ [ 16* 2.352 / 425.32]  ≈   ± 0.297

Here is the graph :  https://www.wolframalpha.com/input/?i=16%2Fn%5E2%C2%A0*+2.352+%3D+425.32

14a^3- 3/a^2=426     [ I'll use "x" instead of "a" ]

14^x^3 -3/x^2 - 426 = 0      multiply through by x^2

14x^5 - 3 - 426x^2 = 0

14x^5 - 426x^2 - 3 = 0

This does not have an algebraic solution, DP   [ or, at least, I don't think that it does ]

There is only one real root at x = a  ≈ 3.12

There are also   4 complex roots

Here is the graph and solution  found by WolframAlpha :

https://www.wolframalpha.com/input/?i=14x%5E5+-+3+-+426x%5E2+%3D+0

We  have the form

y = a(x - h)^2 + k

Where   (h, k) = the vertex = ( -1, 6).....and we know a point on the graph... (0, -1)

So....we can solve for a

-1 = a(-1 - (0))^2  + 6

-1 = a( -1)^2 + 6

-1 = a + 6      subtract 6 from both sides

-7 = a

So....in vertex form, we have

y = - 7( x - (-1) )^2 + 6

y = -7 ( x + 1)^2 + 6