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3 * 4 = \(\boxed{12}\)

This is true because

Red - Pattern 1 or 2 or 3 or 4                      So 4 design

Green - Pattern 1 or 2 or 3 or 4                   So 4 design

Blue - Pattern 1 or 2 or 3 or 4                      So 4 design

So 4 + 4 + 4 = 12

(3,2)

The line can be written as    y = (1/3)(x + 1)     (1)

We need to find a perpendicular line to this one passing through (3,2)

This line will have a slope of  -3.....so we have

y = -3 ( x -3) + 2

y = -3x + 11        (2)

Set (1) and (2) equal to find the  x coordinate of the  point on the circle where these lines intersect

(1/3)(x + 1)  =  -3x + 11             multiply through by 3

x + 1  = -9x + 33      rearrange as

10x = 32

x = 32/10 =  16/5

And y  = -3(16/5) + 11  =   -48/5 + 55/5  =  7/5

So....the intersection is   (16/5, 7/5)

And the distance between this point   and (3,2)  will be the radius of the circle...so....we can find r^2  as

(3 - 16/5)^2 + (2 - 7/5)^2  =

(-1/5)^2 + (3/5)^2 =

1/25 + 9/25  =

10/25  =   2/5   = r^2

So....the area of the circle  =  pi * r^2  =    (2/5) pi units^2

Here's a graph : https://www.desmos.com/calculator/tklpk6vonx

f(g(16))........means that we put 16 into g, first.....and evaluate....

Then...we put that result into f

So

g(16)  =  sqrt (16)  = 4

And

f(4)  = 3 - g(16)  =   3 - sqrt (16)  =  3 - 4  =   -1

The major axis will lie parallel to the y axis

So we have this form   :  (y - k)^2   +  (x - h)^2

                                       _______       _______   =    1

                                          a^2                b^2

The center of the ellipse  is  (2, - 3)  = (h, k)

Since the ellipse is tangent to the y axis and (2, -3) is the center.....then the minor axis must be 4 units in length

So  b = 4/2  = 2   and b^2  = 2

And we can find "a" thusly :   a^2 - b^2  = c^2

a^2  = ???

b^2 = 4

c^2 = (√5)^2  = 5

So

a^2 - 4 = 5

a^2  = 9

a = 3

And the major axis  = 2a  = 2(3)  = 6

Here's a graph : https://www.desmos.com/calculator/iagepa6zop

Here's one way of solving this.....

A perpendicular line will have a slope of (-1/3)

And the equation of this line will be   y  = (-1/3)x      (1)

And we can construct a circle with a radius of 2   centered at the origin.....the equation of this circle will be

x^2 + y^2  = 4     (2)

Sub(1) into (2)  and we have that

x^2 + (-1/3x)^2  = 4

x^2 + 1/9 x^2  = 4

(10/9)x^2 = 4

x^2  = 36/10

x = 6/√10           and y  =  (-1/3)(6/√10) =  -(2/√10)

So....the point  (6/√10, -2/√10)   will lie on the line with a slope of 3

And the equation of this line is

y = 3(x - 6/√10) - 2/√10

y = 3x - 20/√10

The y intercept of this line    (0, -20/√10)

The x intercept of this line  =  (20 / [ 3√10]

And the area of the triangle formed  = (1/2) [20/√10 ] [ 20/ [3√10 ] ]  =  400/60  =  20/3 units^2

Here's the graph : https://www.desmos.com/calculator/axezhbz2qb

The nested functions at the end mean that you put 16 into the function g, then take what you get and put it in the t function (sometimes written as (t o g)(x) )

You can rewrite function t as t(x)=3-√x, because the g function inside of it means that you subrtact the g of x from 3

This is called 'solving by the graphical method'..... when you graph the two (or more ) equations......The points where they intersect is the solution set.

y  =  4 cos (x) + 1

y  = -cos (x) - 4              set the y's equal.....and we have

4cos (x) + 1  = -cos(x) - 4         add  cos (x), 4 to both sides

5cos (x) + 5  =  0        subtract  5 from both sides

5 cos (x)  = - 5              divide both sides by 5

cos (x)  =  -1

And this happens when x  =  pi    and x  = -pi

The intersection point(s) are

4cos (pi) + 1  =  -4 + 1  =  -3   ⇒     (pi, -3)         and

4cos(-pi) + 1  =  -4 + 1  =  - 3  ⇒  (-pi, -3)

See the graph here, GM : https://www.desmos.com/calculator/j9yashts7t

               Midline            Amplitude           Range          

10.         y = 0                    3                      [-3, 3 ]

11.        y  = 0                    4                     [ -4, 4]

12.         y = 0                    2                     [-2, 2 ]

13.        y = 0                     5                     [-5, 5 ]

Option A.....we have the following possible outcomes

(1,1)   (1, 5)     (4,5)

(1,4)    (5, 1)    (5,4)

(4,1)     (4, 4)   (5, 5)

4 of the 9 outcomes have a sum > 6....and 5 don't

So....expected value  =  (4/9)(8) - (5/9)(2)  ≈ $2.44

Option B

We have 8 possible outcomes.....only 3 of them have heads appearing once

So....expected value  =  (3/8)(6)- (5/8)(1)  ≈ $1.63

"B" is better by ≈  $[ 2.44 - 1.63 ]  =  81 cents

Here's my attempt....

We only need to worry about 3 cases.....for.....if 3 or more of the integers are multiples of 3, we can select any three of those to have a sum which is a multiple of 3.....

Case 1.....Two of the integers are multiples of three .....let  R2 = (an integer with a remainder of 2 when divided by 3) and R1 = (an integer with a remainder of 1 when divided by 3 )

So  we have the following possibilities for the other three integers 

R2 R2 R2    just select these three......

R2 R2 R1     select one of the R2's, the R1  and one of the other integers that is a multiple of 3

R1 R1 R2     select the R2, one of the R1's  and one of the other integers that is a muliple of 3

R1 R1 R1     select these 

Case 2   ....Only one integer is a multiple of 3...so....we have these posiibilities for the other four

R2 R2 R2 R2      select any 3 of these 4

R1 R2 R2 R2       select the 3 R2's

R! R1 R2 R2         select one of the R1's, one of the R2's  and the other multiple of 3

R1 R1 R1 R2       select all 3 R1's

R1 R1 R1 R1        select any 3 of these 4

Case 3....No integer is a multiple of 3.....we have

R2 R2 R2 R2 R2     select any 3 of these

R2 R2 R2 R2 R1     select any 3 of 4 R2's

R2 R2 R2 R1 R1     select all the R2's

All the remaining situations will have at least 3R1's....so.....we can guarantee a sum that is a multiple of 3 by selecting from them

Using the properity of geometric equations that the ratio between terms is constant, you can set up the following eqation using the ratios between terms:

\(\frac{x^2}{9x}=\frac{2xy}{x^2} \Rightarrow x^{4}=18x^2y \Rightarrow x^{2}=18y \Rightarrow y=\frac{x^2}{18}\)

Creating another equation so you can solve gets this:

\(\frac{2xy}{x^2}=\frac{8y}{2xy} \Rightarrow 4x^2y^2=8x^2y \Rightarrow 4y^2=8y\Rightarrow y^2=4y\)

Remembering that an equation with a squared term has two solutions, we get y=4 and y=0

Plugging these values into the first equation yields x=0 and x=\(6\sqrt{2} \)

Thus, the first two terms of the sequence are \(54\sqrt{2}\) and 72 or 0 and 0

From here you should be able to find the common ratio R so the 15th term would be \(54\sqrt{2} \cdot R^{15-1}\) or 0

Thank you! 

Well....think about this....if we cut a pie in 13 equal pieces......which would you rather have.....3 of the pieces, or 5 of the pieces????

This hyperbola will have its center at (0,0)

Since  (√13, 0)  and (-√13,0) are the foci....the hyperbola will have the form   

x^2         y^2

___  -   ____  =   1

a^2         b^2

Since  the difference  in  distances from any point on the hyperbola to the focal points  = 6 , we have this property that

2a  =  6       divide both sides by  2

a =  3

a^2  = 9

And     a^2 + b^2  = c^2          where c = √13

So

3^2 + b^2  = (√13)^2

9 + b^2  = 13

b^2 = 13 - 9

b^2  = 4

So....the correct  form is

x^2        y^2

___  -    ___     =  1

 9            4

C'mon....you can do this....

1.5 + x = .25 (15+x)

1.5 + x = 3.75 + .25x 

.75x = 2.25

x = 3

Wel...I don't know what a "box problem," is....but....we can still solve this

Let the amount invested at 5%  = A

Let the amount invested at 9%  = 8000 - A

So.....using   

Principal * interstest rate  = interest earned, we have that

5% * A + 9% * ( 8000 - A)  =  620

.05A + .09 (8000 - A)  = 620        

.05A  + 720 - .09A  = 620

-.04A + 720 = 620        subtract  720 from each side

-.04A  = -100       divide both sides by -.04

A = $2500  =  the amount at 5%

Which means that the amount at 9%  = 8000 - 2500 = $5500

For how many years is that interest?

I will assume 1 year

x = 5% amount

8000 - x = 9% amount

.05 x  + .09 (8000-x ) 620

.05x - .09 x + 720 = 620

-.04x = -100

x = $ 2500  in 5% account       the remainder     8000-2500 = $ 5500 = amount in 9% account

The graph of an inconsisent system - assuming two variables - wiil be parallel lines

For instance

6x + 5y  =  13

6x + 5y  = 2  ......will look like this :  https://www.desmos.com/calculator/nmcyxdjzgm

You need to use the formula which is c=  pi*d

Then plug in which would be C= Pi*3.24

then multiply to get the circumference. What did you get?

Put in you should get \(x = 3\)

\(1.5 + x = .25(15+x) \)

HOW????? the website calculator gave me 9. something