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We want t by itself

n  =  t / p         multiply both sides by p

pn  =  t

You should committ these to memory, IED....they occur quite frequently

6 - 8 - 10

5 - 12 - 13

8 - 15 - 17          are all right triangles

7 - 24 - 26  isn't right

Well, the side lengths 6,8,10 are a right triangle, so, yeah.....

To find any term we can write

a_n  =  -114 + 5 (n - 1)

However....simplifying this we get

an =  -114 + 5n - 5

an =  - 119  + 5n   =    5n - 119

2.

T  =  11cos ( pi * t  / 12 )  + 71

In this case....t  = 2  [ 2 hours after 2 PM ].......so we have

T  =  11 cos (pi * 2 /12)  + 71  =

11cos (pi/6)  +  71  =

11 [ √3/ 2 ]  +  71  =

(11/2)√3   + 71   =

[ 11√3  + 142 ]  / 2   ≈  80.53°

Note that this temperature  will occur again at  -pi/6   because  cos (pi/6)  = cos (-pi/6)

So......solving this for t...we have 

pi * t /12  =  -pi / 6

t    / 12  =  - 1 / 6       multiply  both sides by 12

t =  -12 / 6   =    - 2....i.e., 2 hours before 2 PM   =  12 noon

This makes sense.......the temp  would increase until 2PM.....and at equal times from 2PM, namely at  12 noon    and at 2 PM.....the temps should be equal

Thanks guest :)

I believe it is " software" that you can purchase and download to your computer and can be used in "Excel" spreadsheet. My understanding is that it is quite expensive!! I believe it may cost several thousand dollars(US) to purchase.....You may google "Monte Carlo Simulation Software" and you will have some idea of what is involved.

Hard to read these functions, but I think we have

e(x)  =  .03x^3 + 2

l (x)  =  .5x + 1

So  ..   ( e - l) =   [ .03x^3 + 2 ]  -  [ .5x + 1 ]  =  .03x^3  - .5x + 1

So

(e - l ) (4)  =   .03(4)^3  - .5(4) +  1  =   .92   ...this is in  hundreds, so it converts to 92 people

So....this means that  4 hours after the park opens, the rate that the number of people is changing is 92 people per hour

Here are the outcomes  and their  frequencies

Outcomes  -  Frequencies

2     -    1              7   -  6          12  -  1

3     -    2              8   -  5

4    -     3              9   -  4

5    -     4             10  -  3

6    -     5             11  -  2

P (  even or multiple of 3 )  =

P( even) + P(multiple of 3)  - P(even and multiple of 3)   =

18 / 36  +  12 / 36  -  4/36  =

26 / 36  =     13/18

Yes, 5%

Add the ten football players'  HR and divide this sum by 10 to get

Mean HR = 71.4 BPM

So, the correct answer is 5%, correct?

An estimated percentage woud be  15/300 x 100% = 5% of the apples are bruised/damaged.

(Out of 12,500 apples , that would be  12500x 5% = 12,500 x .05 = 625 apples damaged or bruised)

Thanks Ginger,

I certainly like the way your working begins but I haven't worked out this Euler's totient yet.

I'll just take your word for it.  :)

You should not be so critical of our guest.

There is nothing wrong with good intuition and sometimes the thought processes are very hard to explain.

(2x^1/4y¹) (4x^-3/4y⁴)  =

[2 * 4] *  [ x^1/4 * x^-3/4 ]  * [ y¹ * y ⁴ ]  =

8  *  [ x ^{1/4 - 3/4} ] *  [  y ^{1 + 4} ]  =

8 * [ x ^-2/4 [ * [ y ⁵ ]  =

8x^-1/2y⁵

P(number  < 2  on first draw)   =  1/10

P ( number > 7 on second draw given that a number < 2  was selected first)  =  3/9

So...the total probability is   (1/10) * ( 3/9)   =   3/90  =  1 / 30

Solve this by setting up a system of modular equations.

$\begin{array}{rcll} n &\equiv& {\color{red}0} \pmod {{\color{green}8}} \\ n &\equiv& {\color{red}1} \pmod {{\color{green}9}} \\ n &\equiv& {\color{red}2} \pmod {{\color{green}10}} \\ \text{Set } m &=& 8\cdot 9\cdot 10 = 720 \\ \end{array}$

The first product zero— included as a formality.

 Eurler totients calculated from non-prime numbers.

$\small{ \begin{array}{l} n = {\color{red}0} \cdot {\color{green}9\cdot 10} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}9 \cdot 10) }^{\varphi({\color{green}8}) -1 } \pmod {{\color{green}8}} ] }_{=\text{modulo inverse }(9\cdot 10) \mod 8 } }_{=(9\cdot 10)^{4-1} \mod {8}} }_{=(9\cdot 10)^{3} \mod {8}} }_{=(90\pmod{8})^{3} \mod {8}} }_{=(2)^{3} \mod {8}} }_{= 0} + {\color{red}1} \cdot {\color{green}8\cdot 10} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}8\cdot 10) }^{\varphi({\color{green}9}) -1} \pmod {{\color{green}9}} ] }_{=\text{modulo inverse } (8\cdot 10) \mod {9}} }_{=(8\cdot 10)^{6-1} \mod {9}} }_{=(8\cdot 10)^{5} \mod {9}} }_{=(80\pmod{9})^{5} \mod {9}} }_{=(8)^{5} \mod {9}} }_{=8} + {\color{red}2} \cdot {\color{green}8\cdot 9} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}8\cdot 9) }^{\varphi({\color{green}10}) -1 } \pmod {{\color{green}10}} ] }_{=\text{modulo inverse } (8\cdot 9) \mod 10 } }_{=(8\cdot 9)^{4-1} \mod { 10}} }_{=(8\cdot 9)^{3} \mod {10}} }_{=(72\pmod{10})^{3} \mod {10}} }_{=(2)^{3} \mod {10}} }_{=8}\\ \\ n = {\color{red}0} \cdot {\color{green}9\cdot 10} \cdot [0] + {\color{red}1} \cdot {\color{green}8\cdot 10} \cdot [8] + {\color{red}2} \cdot {\color{green}8\cdot 9} \cdot [8] \\ n = 0+ 640 + 1152 \\ n = 1792 \\\\ n \pmod {m}\\ = 1792 \pmod {720} \\ = 352 \\ \mathbf{n_{min}} \mathbf{=} \mathbf{352} \end{array} }$

I think it is

"Assign each taxpayer in town a number and then randomly select 500 of these numbers."

(x - 5) (x + 5)  < 0

x^2  - 25  < 0

Note that   (-5)   makes   (-5)^2 - 25  < 0    ⇒  25 - 25 < 0  ⇒  0 < 0  false

But

-4  makes   (-4)^2 - 25  < 0 ⇒   16 - 25 < 0  ⇒  -9 < 0    true

So...

x  = -4 is the smallest integer that makes this true

See this graph : https://www.desmos.com/calculator/6iwm6iiww6

(f o g)(4)   =   f( g(4) )

g(x)   =   -2x - 6

g(4)   =   -2(4) - 6

g(4)   =   -8 - 6

g(4)   =   -14

(f o g)(4)   =   f( g(4) )   =   f( -14 )

f(x)   =   3x - 7

f( -14 )   =   3(-14) - 7

f( -14 )   =   -42 - 7

f( -14 )   =   -49

(f o g)(4)   =   f( g(4) )   =   f( -14 )   =   -49

Is this what you mean?

∑[2n -1], n=-1 to 15 = 221

f(x)  =  -3(x)^4  + 7x^2

If even....f(-x)   = f(x)

f(-x)   =  -3(-x)^4  + 7(-x)^2    =   -3(x)^4 + 7(x)^2   which is true

To show that it is odd, we must have that   -f(x)  =   f(x)  .....so....

-f(x)  =   -  [ -3x^4 + 7x^2  ]  =  3x^4  - 7x^2   ....not equal to f(x)

Thus...it is even

P.S.     Even functions are symmetric about the y axis

Look at the graph, here :   https://www.desmos.com/calculator/uooawffq1f

The slope of this line is   ( 7 - -3) / ( 1 - -2)   =  10/ 3

Using either point to write the equation, we have

y =  (10/3) ( x - 1) + 7      multiply both sides by 3

3y  = 10 (x - 1)  + 21

3y  = 10x - 10 + 21

3y =  10x + 11     rearrange as

10x - 3y  =  -11

N(O(N(O(N(O(3))))))

O(3)  =  9

N(O(3)) = N(9)  = 6

O(6)  =  36

N(O(6))  = N(36) =   12

O (12)  =  144

N(O(12)))  =  N(144)  =   36