All Answers 352712 Answers

1  15  105  455  1365  3003  5005  6435  6435  5005  3003  1365  455  105  15  1  = R 15
Row 15th has 105 as the 3rd combination. But, as Anthrax says, the 2nd combination in the 100th row is 100 and it begins like this:
1  100  4950  161700  3921225  75287520  1192052400  1 6007560800  18 6087894300  190 2231808400  .........etc,

Find the value of   \(\sin40°\cdot\cos50°+\cos40°\cdot\sin50°\) .

Remember the angle addition formula for sine:    \(\sin(\alpha+\beta)\ =\ \sin\alpha\cos\beta+\cos\alpha\sin\beta\)

So...

\(\sin40°\cdot\cos50°+\cos40°\cdot\sin50°\ =\ \sin(40°+50°)\ =\ \sin(90°)\ =\ 1\)

_ _ _ _ _

\(\text{An earlier similar problem established that the roots are }\\ r = \sqrt{2}+\sqrt{5},~\sqrt{2}-\sqrt{5},~-\sqrt{2}+\sqrt{5},~-\sqrt{2}-\sqrt{5}\\~\\ \text{which results in }\\~\\ p(x) = x^4-14 x^2+9\\ p(1) = 1-14+9 = -4\)

In \(\triangle PQR,\) we have \(\angle P = 90^\circ,\) \(PQ=3,\) and \(QR=\sqrt {58}.\) Find \(\tan R.\)

By the Pythagorean theorem,

3² + ( PR )²  =  √[ 58 ]²

9  +  ( PR )²  =  58

( PR )²  =  49

PR  =  7

tan( angle )  =  opposite / adjacent

tan( R )  =  3 / PR

tan( R )  =  3 / 7

Find AC in the diagram to the nearest integer.

sin( angle )  =  opposite / hypotenuse

sin( 70° )  =  31 / AC

                                         Multiply both sides of the equation by  AC

AC sin( 70° )  =  31

                                         Divide both sides of the equation by  sin( 70° )

AC  =  31 / sin( 70° )

                                         Plug    31 / sin( 70° )    into a calculator.

AC  ≈  33

Some say that we are living in a simulation...

and that is why EVERYTHING works in a mathematical way.

The answer would be the 3rd option since 2 is 20% of the 10 coins being selected.

Hope this helps.

Examples:

To enter:     \(\frac12\)  ,     type:     1/2

To enter:     \(\frac{15}{16}\)  ,     type:     15/16

I believe you can thank the writers at Britannica.com for that.

I wrote a relatively short computer code to solve them for you. The code uses the Chinese Remainder Theorem + Modular Multiplicative Inverse. One of the mathematicians here is much better suited in explaining it to you than I am.

i=1;a1=5;a2=7;a3=9;a4=11;a5=1;a=((a1*a2*a3*a4*a5)/a1);m=a1;a=a%m; if(a*i % m==1,goto loop, goto loop1);loop:print"The mmi = ",i;loop1:i++;if(i<=(m-1), goto3, discard=0;
r1=1;r2=2;r3=3;r4=4;r5=0;listforeach(x, m=product(a1, a2, a3, a4, a5),(( n=m/a1*r1*2+ m/a2*r2*3+ m/a3*r3*4 + m/a4*r4*8 + m/a5*r5*0));printm,"m + ",n%m;return

OUTPUT:   3465m + 1731, m=0, 1, 2, 3........etc.

Nvm, the last one I got it. It is 720

Ok Penquino, see you later.

Have fun :)

Looks like 126 to me.

Simplify first equation:

2x - 3y  =  -3.2 - 0.2x + 0.1y

                                                Add  0.2x  to both sides of the equation.

2.2x - 3y  =  -3.2 + 0.1y

                                                Add  3y  to both sides.

2.2x  = 3.1y  - 3.2

Simplify second equation:

x  =  0.6x - y + 8.8

                                      Add  y  to both sides of the equation.

x + y  =  0.6x + 8.8

                                      Subtract  x  from both sides.

y  =  -0.4x + 8.8

Substitute  -0.4x + 8.8  in for  y  in the first equation and solve for  x .

2.2x  =  3.1( -0.4x + 8.8 ) - 3.2

                                                     Distribute  3.1  to the terms in parenthesees.

2.2x  =  -1.24x + 27.28 - 3.2

                                                     Add  1.24x  to both sides.

2.2x + 1.24x =  27.28 - 3.2

                                                     Combine like terms.

3.44x  =  24.08

                                                     Divide both sides by  3.44

x  =  7

Substitute  7  in for  x  in the second equation to find  y .

y  =  -0.4( 7 ) + 8.8

y  =  6

sin 30 = 1/2

cos 60 = 1/2

tan 45 = 1

1/2 * 1/2 * 1 = 1/4

Gather like terms in the two equations to get :

2.2x - 3.1y = -3.2               (edited here due to error)

.4x+y = 8.8             Multily this equation by 3.1

1.24 +3.1 y = 27.28     and add this to the first equation to get

3.44x = 24.08      x= 7 

Sub this in to one of the original equations to find y = 6

I agree with the other answer, this question is a tad confusing.

If we interpret it as each number being a number instead (weird sentence, I know), 100 would actually be the smallest three-digit number in Pascal's triangle. It is the second number in the 99th row (or 100th, depending on who you ask), or \(\binom{100}{1}\)

This is quite eloquent, but I think this exceeds the word count restriction. 

There are countless reasons why math exists. It's the key to how life functions.

This video perfectly explains why math exists: https://www.pbs.org/wgbh/nova/video/the-great-math-mystery#.Vx0zZeAY4jM.email

1 < 2x < 2      divide through by 2

1/2 < x < 1

Also

1 < 3x < 2      divide through by 3

1/3 < x < 2/3

We need the most restrictive interval that solves both of these inequalities....this is  (1/2 , 2/3)

In effect ...he only makes  10 - 3  = $7  for each pizza delivered

So....the number of pizzas  =   5000/7  ≈  715  pizzas

g^4 + 12g^2 + 9       complete the square

g^4 + 12g^2 + 36 + 9 - 36       simplify

(g^2 + 6)^2  -  27

q  =  - 27

Thank you!

x^2 + 5x + 8  = 0      complete the square on x

x^2 + 5x + 25/4  = -8 + 25/4

(x + 5/2)^2  = -32/4 + 25/4

(x + 5/2)^2  =  -7/4     take both roots

(x + 5/2)  =  ± [ √7  / 2 ] i   subtract 5/2 from both sides

x =   -5/2 ±  [√7 / 2 ] i

a =-5/2  b^2  = 7/4

So

a + b^2  =  -5/2 + 7/4  =   -10/4 + 7/4  =   -3/4

OK, penquino.....have a good Summer  !!!!