All Answers 356001 Answers

We do like people to show appreciation, that is true.

And I thank you for yours.

HOWEVER

We also want to help people learn. When people ask a lot of questions in a short period of time as you have done it is fairly clear to me that you are primarily using this site as a cheat site. You are getting our well meaning answerers to do your homework for you.

This assessment is backed up by your not asking ANY of the answerers any questions at all about their solutions. You have not interacted with them in any way.

It depresses me when this site is used/abused like this.

I do not think that you have posted the whole question.

This is presumably a geometry question. But the figure is missing, and we don't know the exact location of point F. 

If no figure is given, the problem cannot be solved.

True!

As follows

Thanks Max, I am really pleased that you have answered this :))

Let a, b, c, d are all prime numbers, and they meet the following guidelines: a>3b>6c>12d and a^2-b^2+c^2-d^2=1749

Please determine all the possible values of a^2+b^2+c^2+d^2.

I got that d=2 but that's about it. 

You have the correct idea.....

The angle that you found is  a reference  angle of   59.5° in the 4th quadrant 

But....If you look carefully, you will  note that the measure of  this angle plus the measure of  θ  = 180°

So....to find θ....we have that

59.5 +  θ  =  180        subtract 59.5 from both sides

θ  = 120.5°

LCM[12, t]^3 =(12t)^2, solve for t

t = 18, so that:

LCM[12, 18]^3 =LCM[1728, 5832] =46,656

46,656 =(12t)^2=144t^2

t^2 =46,656 /144 =324

t = sqrt(324) = 18

Yes, it's pretty neat!

Nicely done, Rom.....!!!!

Here's another way to see this

P(at least two match)  =     1 - P(that none match)

Note....that for none to match....the first die has 6/6 a chance of showing any number.....the second has a 5/6 chance of not showing the first number...and the third die has a 4/6 chance of not showing either of the first two numbers

So....P(none to match)  =  (6/6)(5/6)(4/6)  = 20/36  = 5/9

So....P (at least two match)  =  1 - 5/9   =  4/9

Let O be the center of the circle.....we will have 5 triangles that comprise [ABCDE ]

Three of these AOB, BOC and DOE will be equilateral triangles with  areas of  [2^2 * sqrt (3) / 4] = √ (3) units^2

So...their combined areas  =   3√ (3) units^2

The arc of the circle covered by the chords  AB, BC and CD will be 180°

So....the chords  CD and EA occupy  90° each

So...triangles COD and EOA  will be be right isosceles triangles each with an area of (1/2)(2)^2 = 2

So....their combined areas =  4 units^2

So  [ ABCDE ] =     [ 4 + 3√3 ]  units^2

Here's a pic :

\(P[\text{at least 2 match}]=\\ 1-P[\text{none match}] = \\ 1-\dfrac{\dbinom{6}{3}3!}{6^3} = \\ 1 -\dfrac 5 9 = \\ \dfrac 4 9\)

Here's another way to see this :

1/x  +  1/y =   1/7

xy = 7 (x + y)

7(x + y) - xy = 0

7x + 7y - xy = 0

7x + y(7 - x) = 0

7x  =  -y (7 - x) 

7x =  y(x - 7)

y =   7x / ( x - 7)

y = 56   when x = 8

y = 14   when x = 14

y = 8 when x = 56

etc.....

\((a+i b)^3 - 107i = (a^3-3 a b^2) + i(3 a^2 b - b^3-107)\\ \text{If this is to be real, then}\\ 3 a^2 b - b^3-107 = 0\\ (3a^2-b^2)b = 107 \)

\(\text{the only factors of 107 are }1,~107 \text{ so either}\\ b=1,~(3a^2-b^2)=107 \text{ or }b=107,~(3a^2 - b^2)=1\)

\(\text{If the first}\\ (3a^2 - b^2) = 3a^2 - 1=107\\ 3a^2 = 108\\ a^2 = 36\\ a=\pm 6\)

\(\text{If the second}\\ (3a^2 - b^2) = 3a^2 - 107^2 = 1\\ 3a^2 = 1 + 107^2 = 11450\\ a^2 = \dfrac{11450}{3}\\ \text{and this doesn't lead to }a \text{ being an integer}\)

\(\text{So our solution is}\\ r = (6+i)\\ (6+i)^3-107i = 198\)

I believe this works  at  10 o'clock   and at  2 o'clock

1/x  +  1/y  = 1/7

x + y  =  xy / 7

7 (x + y ) = xy

7 =   xy / ( x + y)

x = 8 , y = 56

x = 14, y = 14   [ if x , y  are not distinct ]

x = 56, y = 8

Sum =  8 + 14 + 56  =   78

Just type

abs (number)      and hit enter

Number of degrees interior angle pentagon =  108

Number of degrees interior angle of a hexagon = 120

So

108 /120 =   9/10

Your answer for the ratio of the  interior angle sums is correct...!!!

I tried that for guest, and I am pretty sure it is incorrect. I think p(x) can equal 8 because if x = 7, p(x) = 7+1 = 8. 

yes can u please help this is very confusing

Thanks, Josuuauy....!!!!

We always like those kinds of compliments   !!!!

We have the form

y = Asin (Bx + C) + D

A= the amplitude = 3

B can be found as follows

B = 2pi / period =  2pi / 4pi   =  1/2

C = phase shift.....there is none..so...C = 0

D = shift up/down.....the midline for the normal sine function would be y = 0

So....this is shifted up 2 units since the midline is y = 2

So....the function is

y = 3sin (  [ 1/2]  x) + 2

Here's the graph : 

The first looks pretty linear to me

y changes by 20  when x changes by 1

So

y = 20x

And the proprtionality constant  (slope) of the data for truck 1  =  20

So  A and C  are correct

For Truck 2.....

Note that the slope is not constant

The slope from hour 1 to hour 2 =  90 - 45  =  45

However.....the slope from hour 2 to hour 3  = 145 - 90  =  55

So.....this is not linear because the slope is not constant

So   E is another correct answer